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Induced coactions along a homomorphism of locally compact quantum groups

We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.


Introduction
Induced coactions of a locally compact group from its closed subgroup on C*-algebras were studied by Rieffel [25], with a motivation coming from Mackey's works on imprimitivity. Induced coactions from compact subgroups play an essential role in the reformulation of the Baum-Connes conjecture by Meyer-Nest [19]. Vaes [32] has developed the theory of induction and imprimitivity for a locally compact quantum group and its closed quantum subgroup. This induction procedure for quantum groups gives a fundamental tool to study the quantum analogue of Baum-Connes conjecture in [36], [37] for example.
For a quantum group G and its closed quantum subgroup H with suitable conditions, the quantum homogeneous space G/H can be given by the induced G-coaction Ind G H C from C with a trivial Hcoaction. In the classical case, a homogeneous space G/H can be also constructed by taking the quotient space of G with the right H-action, or equivalently by taking the fixed point algebra of C 0 (G) with the right H-coaction. When the quantum subgroup H is compact, this method still works as shown in [28]. However, in the general non-compact case, we have technical subtleties to construct C r 0 (G/H) as a fixed point algebra. There is difficulty in finding a good notion of "properness" for coactions on C*-algebras to integrate sufficiently many elements of a C*-algebra by the Haar measure, which can be an infinite measure. See [26], [8] for some approaches to construct fixed point algebras.
Another interpretation for induced coactions and fixed point algebras is to regard them as (the coordinate rings of) balanced products of non-commutative spaces with quantum group actions, which should be realized as C*-algebraic cotensor products if exist. For C*-algebras with coactions of a compact quantum group, such a construction has appeared in [10], [36]. Again when the quantum group is noncompact, it becomes much harder to find appropriate constructions for cotensor products of C*-algebras.
In this paper, we give a unified approach to induced coactions for a closed quantum subgroup and fixed point algebras. We introduce a generalized concept of induced coactions along a general homomorphism of locally compact quantum groups, in a similar spirit as in [14]. Next, we introduce a class of homomorphisms with a measure theoretic condition which behaves particularly nicely with our framework. This class of homomorphisms has appeared in [15] in a different formulation, and classically it corresponds to the class of continuous homomorphisms of locally compact groups that are proper as continuous maps of topological spaces. We would like to call such a homomorphism a proper homomorphism in this paper. We construct induced coactions along a proper homomorphism. We do this by decomposing the homomorphism into a closed quantum subgroup and the Pontryagin dual of an open quantum subgroup. Moreover, we use this decomposition to investigate the properties of induced coactions. Thus our work heavily depends on the techniques developed in [32], where induced coactions along closed quantum subgroup were constructed, and those in [12], where the concept of open quantum subgroups was investigated. Also, we should mention that many motivating examples and ideas in this paper come from [23].
For a locally compact quantum group G, we have a new quantum group called the quantum double of G, which contains G as a closed quantum subgroup. It is observed in [23] that the procedure of taking induced coactions from a closed quantum subgroup H of G is compatible with the coactions of their quantum doubles D(G) and D(H). More precisely, if the C*-algebra A has an H-coaction which extends to a D(H)-coaction, then this extension induces a D(G)-coaction on the induced G-coaction Ind G H A. In this paper, we show slightly stronger statement holds for more general settings. This can be seen as a comodule analogue of base change formula for modules over algebras, and it can be stated as follows if we could use cotensor products (see Theorem 5.3 for the precise statement). Consider a commutative diagram of locally compact quantum groups and homomorphisms, Then for a C*-algebra A with an H-coaction, it holds C r 0 (G) A, if this holds when A = C. Although its conclusion looks quite intuitive, the verification involves several technically naive points. Practically, we consider the situation when E and F are closed quantum subgroups of G and H, respectively. Then it will give a good tool for calculation which reduces the induction procedure to smaller quantum groups. As already mentioned, a quantum homogeneous space G/H has two (candidates of) interpretations. One is an induced coaction of C along H → G, and the other is a fixed point algebra of C r 0 (G) with the right H-coaction. Our approach of induced coactions gives a way to compare them when we formulate the latter interpretation as an induced coaction of C r 0 (G) along a trivial homomorphism H → 1, where 1 is the trivial group. We give a criterion for their coincidence, and we will also see this coincidence gives a sufficient condition for the case A = C of the analogue of base change formula above.
For monoidally equivalent two compact quantum groups, it is proved that various properties are preserved in [10], [36] by using cotensor products with the biGalois object. It can be hoped that the techniques of the generalized induced coactions will give some tool to compare two quantum groups which need not be monoidally equivalent but still have certain similarities. As an application, we consider a certain problem of recovering a locally compact quantum group from its category of a certain kind of G-comodules with the forgetful functor, in a similar spirit with Tannaka-Krein reconstruction theorem. For compact quantum groups, a celebrated result was obtained by Woronowicz [38]. But if we consider possibly non-compact cases, the situation becomes significantly difficult, although Tatsuuma managed to give a reconstruction theorem in [30] for locally compact groups.
On the other hand, there is another approach taken by [20], as an answer for a question by Debashish Goswami. Instead of the category Rep(G) of unitary representations, they consider the category C G of G-equivariant C*-algebras with G-equivariant morphisms of C*-algebras for a locally compact quantum group G (or more generally a C*-bialgebra associated with a manageable multiplicative unitary). Then they show that there is a bijective correspondence of a homomorphism H → G of locally compact quantum groups and a functor C G → C H which commutes with the forgetful functors C G → C 1 ← C H . See [20,Theorem 6.1] for detail. We consider similar situations for the 1-categories of coactions with two different kinds of morphisms. One is the 1-category of isomorphism classes of equivariant correspondences Corr G , and the other is the equivariant Kasparov category KK G . For C = Corr or KK and a certain pair (G, H) of locally compact quantum groups, we will construct a categorical equivalence C G ∼ − → C H that commutes with the forgetful functors C G → C 1 ← C H up to a natural isomorphism in Corollary 7.5. We also see even when G, H are non-monoidally equivalent finite quantum groups, they can form such a pair.
Here we should note that our situation is slightly weaker than that considered in [20], where they impose that the commutativity with forgetful functors must be equality, not up to a natural isomorphism.
We explain the organization of this paper. In Section 2, we recall the basic concepts about locally compact quantum groups defined by [16]. We also fix the notation. In Section 3, we define the formal generalization of induced coactions from closed quantum subgroups. We prove few properties including induction in stages (Theorem 3.8). We also discuss the relationship between two interpretations of quantum homogeneous spaces in Section 3.2. From Section 4, we focus on the case of a proper homomorphism φ : H → G of locally compact quantum groups. In Section 4.1, we show the existence of induced coactions along φ when G, H are regular, as a consequence of induction in stages. As a supplementary topic, we give another description of induced coactions when H = K is compact in Section 4.2. This will justify the interpretation of induced coactions as cotensor products for the compact case. In Section 5, we prove the analogue of base change formula for induced coactions (Theorem 5.3). We also give some concrete situations of this theorem by using the terminologies from Appendix A about double crossed products and Yetter-Drinfeld C*-algebras. For completeness in Section 6, we show several properties of induced coactions along a proper homomorphism φ that were previously known in the case of closed quantum subgroups by [32], [23]. We emphasize the functorial aspects of induced coactions. We consider a relationship of induced coactions and twisted tensor products (Proposition 6.3), imprimitivity for reduced crossed products (Proposition 6.4), and adjointness of induction and restriction (Proposition 6.6). Finally in Section 7, we use these techniques to compare categories of coactions of certain double crossed products with a discrete abelian group. Especially, we show there is a pair of non-isomorphic finite quantum groups whose categories of coactions are equivalent in a compatible way with forgetful functors (Corollary 7.5).

Preliminaries
In this section, we recall and fix notation and terminologies about quantum groups and their coactions on C*-algebras. We roughly follow [29] and [17] for general notation about operator algebras and Hilbert C*-modules.
Let A, B be C*-algebras. For a non-degenerate c.p. map f : A → M(B), we abuse notation and simply write f : M(A) → M(B) for M(f ) : M(A) → M(B) which is the unique u.c.p. map extending f such that it is strictly continuous on the unit ball of M(A). We also write 1 A for the unit of the multiplier C*-algebra M(A). Unless stated otherwise, A ⊗ B always means the spatial tensor product of A and B. We will use⊗ for von Neumann algebraic tensor products. For C*-algebras A, B and Hilbert spaces H, H ′ , we write σ and Σ for the flipping maps σ : A ⊗ B ∼ = B ⊗ A and Σ : H ⊗ H ′ ∼ = H ′ ⊗ H, respectively. We also use σ for the multipliers of tensor products of C*-algebras and the tensor products of von Neumann algebras. We also use the so-called leg-numbering notation. By using this we can write σ = (−) 21 . Often we ignore the leg of C.
When X, Y are sets and f, g : X → Y are maps, we write Ker(f, g) When Y is a subset of a Banach space X, we write [Y ] for the norm-closure of the linear span of Y in X.
Definition 2.1. Let A be a C*-algebra and ∆ : A → M(A ⊗ A) be an injective non-degenerate *homomorphism.
(  ( Let G = (L ∞ (G), ∆ G ) be a locally compact quantum group. The left or right Haar weight is unique up to a scalar multiplication. Take the left Haar weight h of G. Via the GNS construction of h, we obtain a Hilbert space L 2 (G) with Λ : {x ∈ L ∞ (G) | h(x * x) < ∞} → L 2 (G), and a normal faithful * -representation M → B(L 2 (G)). Moreover we can obtain a multiplicative unitary for any x, y ∈ Dom(Λ), where Λ ⊗ Λ means the map given by the GNS construction of h ⊗ h. It holds ∆ G = Ad W G * (−) 2 . We shall write K(G) = K(L 2 (G)) for short, and often identify K(G) * as B(L 2 (G)) * .
We write C r 0 (G) : , which is a well-defined C*-algebra with C r 0 (G) ′′ = L ∞ (G). We still write ∆ G for the restriction of ∆ G on C r 0 (G), and it holds (C r 0 (G), ∆ G ) is a well-defined bisimplifiable C*-bialgebra.
There is a locally compact quantum group G, which we call the dual of G, such that the GNS construction of the left Haar weight of G can be canonically identified with L 2 (G) and W G = ΣW G * Σ under this identification. We write W G := W G . When we write ∆ cop G := σ∆ G , then G op := (L ∞ (G), ∆ cop G ) and G cop := (L ∞ (G) op , ∆ G ) are also locally compact quantum groups, where L ∞ (G) op means the opposite algebra of L ∞ (G). We write J G for the modular conjugation of the left Haar weight of G. Then we have an isomorphism of von Neumann bialgebras For a locally compact quantum group G we also have the universal version (C u 0 (G), ∆ u G ) of the C*bialgebra (C r 0 (G), ∆ G ), and a canonical homomorphism of C*-bialgebras λ G : Definition 2.3. Let G be a locally compact quantum group.
(1) We say G is a compact quantum group if C r 0 (G) is unital.
When a locally compact quantum group G is regular, then G and G op are also regular, and we have ) with respect to the right regular representation of G. This satisfies ∆ G = Ad V G (−) 1 . We warn that in some references the symbol W is used instead of V , which roughly corresponds to the different choice of left or right for the GNS construction of Haar weights.

Homomorphisms and bicharacters.
We recall several forms of homomorphisms between quantum groups by following [20]. Definition 2.4. Let G, H be locally compact quantum groups.
(2) A homomorphism of locally compact quantum groups from H to G is a homomorphism of C*-bialgebras C u 0 (G) → M(C u 0 (H)). We use a formal symbol φ : H → G to indicate a homomorphism from H to G, and we write φ u : C u 0 (G) → M(C u 0 (H)) for the homomorphism of C*-bialgebras associated with φ.
(3) For a homomorphism φ : H → G, we define φ r : C r 0 (G) → M(C r 0 (H)) as a * -homomorphism with φ r λ G = λ H φ u , if it exists. Such a * -homomorphism is unique if it exists, and φ r is necessarily a homomorphism of C*-bialgebras.
Let G, H be locally compact quantum groups. For a homomorphism φ : H → G, it holds that there is the non-degenerate * -homomorphism φ * ∆ G : It also holds that φ * ∆ G = Ad W φ * (−) 2 . The correspondence φ → W φ is bijective as a map from the set of homomorphisms H → G to the set of bicharacters in UM(C r 0 (H) ⊗ C r 0 ( G)). Any homomorphism C r 0 (G) → M(C r 0 (H)) of C*-bialgebras has the form of φ r for some homomorphism φ : H → G, and we have For a homomorphism φ : H → G, there are well-defined homomorphisms φ : G → H and φ op : When φ : H → G and ψ : G → F are homomorphisms of locally compact quantum groups, their composition ψφ : H → F can be given by (ψφ) u := φ u ψ u . This is a well-defined homomorphism of locally compact quantum groups and the corresponding bicharacter W ψφ satisfies W ψ 23 W φ 12 = W φ 12 W ψφ 13 W ψ 23 . We have the trivial homomorphism 1 H→G : H → G determined by the bicharacter 1 ∈ UM(C r 0 (H) ⊗ C r 0 ( G)). The identity homomorphism id G : G → G which is given by

2.3.
Coactions. Next, we collect definitions about coactions on C*-algebras of quantum groups.
Definition 2.5. Let G be a locally compact quantum group, and A be a C*-algebra. Let α : homomorphism. When f is non-degenerate, f is a left G- * -homomorphism if it holds that βf = (id ⊗f )α. When α is continuous, we say a * -homomorphism f : A → M(B) (which need not be non-degenerate) is a left G- * -homomorphism if for all a ∈ A and x ∈ C r 0 (G), it holds that (x ⊗ 1 A )(βf (a)) = (id ⊗f )((x ⊗ 1 A )α(a)). (5) Consider a von Neumann algebra M and an injective normal unital * -homomorphism α : M → L ∞ (G)⊗M . We say α is a von Neumann algebraic left G-coaction on M if (id⊗α)α = (∆ G⊗ id)α. We define a right G-coaction α : A → M(A ⊗ C r 0 (G)) on a C*-algebra A so that σα is a left G opcoaction on A, and a von Neumann algebraic right coaction in a similar way. Likewise, we also define continuity and a G- * -homomorphism for a right coaction.
For a locally compact quantum group G, a left G-coaction is continuous in the weak sense if it is continuous in the strong sense, and when G is regular the converse also holds by [4,Proposition 5.8].
For a locally compact quantum group G and a left G-C*-algebra (A, α), we can consider α(A) as a left G-C*-algebra with ∆ G ⊗ id A , and α : A → α(A) is a left G- * -isomorphism. When C is a C*-algebra (with a trivial coaction), A ⊗ C is a left G-C*-algebra with α ⊗ id. Also, C r 0 (G) can be considered as a left G-C*-algebra with ∆ G and as a right G-C*-algebra again with ∆ G . Unless stated otherwise, we always equip these algebras with the coactions described here.
For a left G-C*-algebra (A, α), we can define the reduced crossed product G ⋉ which is a C*-algebra with the continuous right G-coaction Ad V G 13 (−) 12 . For a homomorphism φ : H → G and a left G-C*-algebra (A, α), there is the left continuous H-coaction φ * α on A such that (id ⊗α)φ * α = (φ * ∆ G ⊗ id)α. We call φ * α the restriction (or pullback) of α along φ. We sometimes write φ * A in this situation to make it clear what quantum group coacts on A. Similarly for a right G-C*-algebra (A, α), we can also define the restriction α * φ of α along φ as the right continuous H-coaction on A which satisfies (α ⊗ id)α * φ = (id ⊗∆ G * φ)α. Also, we sometimes write A * φ. Now we recall and fix terminologies about coactions on Hilbert C*-modules and C*-correspondences. We refer [2] for the details of the definitions and the constructions. Consider a locally compact quantum group G and left G-C*-algebras (A, α), (B, β). For a right B-Hilbert module E, we can define a left or right coaction and its continuity in a similar way as in Definition 2.5, but we additionally require [κ(E)(C r 0 (G) ⊗ 1 B )] = C r 0 (G) ⊗ E for a left G-coaction κ of E to be continuous. A left G-coaction κ on E gives rise to the left coaction on the linking algebra K B (E ⊕ B) whose restrictions to B and E equal β and κ, respectively. By abusing notation we continue to write κ for this coaction on K B (E ⊕ B). It is shown in [32,Remark 12.5] that when G is a regular locally compact quantum group and (B, β) is a left G-C*-algebra, a left G-coaction κ on a right Hilbert B-module E is automatically continuous, whose corresponding left coaction on K B (E ⊕ B) is also continuous.
For an (A, B)-correspondence (E, π) (i.e. a right Hilbert B-module with a * -homomorphism π : In this case we say (E, π) is a left G-(A, B)correspondence. The inner tensor product of left G-C*-correspondences has a canonical structure of a left G-C*-correspondence. We say a left G-(A, B)-correspondence E is a left G-(A, B)-imprimitivity bimodule if E is also an imprimitivity bimodule. In such a situation, A and B are said to be left G-Morita equivalent.

2.4.
Closed and open quantum subgroups. We recall the notions of closed quantum subgroups and open quantum subgroups by following [9], [12].
Definition 2.6. Let G, H be locally compact quantum groups and φ : H → G be a homomorphism.
(1) φ gives a closed quantum subgroup if φ r exists and φ r extends to a normal injective unital *homomorphism L ∞ ( H) → L ∞ ( G). We still write φ r for this extension. (2) φ gives an open quantum subgroup if φ r exists and φ r extends to a normal surjective unital * -homomorphism L ∞ (G) → L ∞ (H). We still write φ r for this extension.
We have followed the definition of a closed quantum subgroup by Vaes. By [9, Theorem 3.5, Theorem 3.6], if φ : H → G gives a closed quantum subgroup then φ u : surjective. An open quantum subgroup is a closed quantum subgroup by [12,Theorem 3.6].
We collect some properties about open quantum subgroups and their consequences (see Section 2 in [12]). Consider a homomorphism ι : H → G giving an open quantum subgroup. Let P = P ι ∈ PZL ∞ (G), where P indicates the set of projections, be the central support of ι r : L ∞ (G) ։ L ∞ (H). Then P satisfies [W G , P ⊗ P ] = 0, and ∆ G (P )(P ⊗ 1) = P ⊗ P = ∆ G (P )(1 ⊗ P ) in B(L 2 (G) ⊗2 ). It follows H = (L ∞ (H), ∆ H ) can be canonically identified as (P L ∞ (G), (P ⊗ P )∆ G (−)), with the left and right Haar weights given by restricting those of G via P . We write p = p ι for P considered as a coisometry L 2 (G) ։ L 2 (H), so we have P = p * p. It holds ι r = p(−)p * : L ∞ (G) → L ∞ (H) = pL ∞ (G)p * , and (p ⊗ p)W G = W H (p ⊗ p). We also remark that the compatibility of the left Haar weights shows pJ G = J H p. The calculations is a well-defined conditional expectation by taking ω ⊗ id for each ω in B(L 2 (G)) * or B(L 2 (H)) * to them. This map also gives the conditional expectations L ∞ ( G) ։ ι r L ∞ ( H) and M(C r 0 ( G)) ։ ι r M(C r 0 ( H)). Finally, we remark that if G is regular, then H is also regular. To see this, since we have

Formal aspects of induced coactions
First, we recall the description in [32] of induced coactions along a homomorphism φ : H → G of regular locally compact quantum groups which gives a closed quantum subgroup. We write the von Neumann algebra L ∞ (G/H) := {x ∈ L ∞ (G) | Ad V φ x 1 = x ⊗ 1}, and when G, H are regular there is a C*-subalgebra C r 0 (G/H) ⊂ L ∞ (G/H). The restrictions of ∆ G give a von Neumann algebraic left G-coaction on L ∞ (G/H) and a continuous left G-coaction on C r 0 (G/H). We note Ad V G 12 (−) 1 gives These * -isomorphisms are from algebras with one leg and algebras with two legs, and it will be convenient to distinguish these algebras for later usages of the leg-numbering notation. Let which is an (L ∞ ( G) ∨ L ∞ (G/H), L ∞ ( H))-imprimitivity W*-bimodule (see the remark after [32,Definition 4.2]). Now suppose G, H are regular. There is I φ ⊂ I φ which is a ([C r 0 ( G)C r 0 (G/H)], C r 0 ( H))imprimitivity (C*-)bimodule by Theorem 6.2 (or more generally Theorem 7.3) and the argument from Theorem 8.2 in [32]. Actually I φ is constructed before C r 0 (G/H), and then C r 0 (G/H) is constructed by using [32,Theorem 6.7], which is the quantum version of Landstad's theorem. It follows that I φ ⊂ I φ is dense with respect to the strong* operator topology of B(L 2 (G) ⊕ L 2 (H)), and that More generally for a left H-C*-algebra (A, α), there exists the C*-subalgebra Ind φ A ⊂ M(C r 0 (G) ⊗ A) with the induced left G-coaction. This is constructed as a certain C*-subalgebra of the C*-algebra that will turn out to be M(G ⋉ r Ind φ A), again by using [32,Theorem 6.7]. Also, by Theorem 7.3 and the proof of Theorem 8.2 in [32], it holds that [I φ We note that the imprimitivity theorem for full crossed products is also shown in [32] and locally compact quantum groups are assumed to be strongly regular. But as long as we concern only with the reduced crossed products, regularity is enough to show the results quoted above. Indeed, it is already pointed out by the remark after Theorem 6.2 in the original paper [32] and by Remark 1.5 in [12], that the existence of the imprimitivity bimodule I φ do not require strong regularity. We can continue to check the proofs of the quoted facts still work only with regularity. We list the results of [32] which were stated under the assumption of strong regularity, that we will use hereafter: Theorem 7.2, Theorem 7.3 for reduced crossed products, the proof of Theorem 8.2. We only need these results for injective coactions.
3.1. Definition and first properties. The left G-C*-algebra Ind φ A above satisfies the characterization due to [32,Theorem 7.2]. In order to consider induced coactions in more general settings, we formally extend that characterization as follows.
Definition 3.2. Let φ : H → G be a homomorphism of locally compact quantum groups, and (A, α) be a C*-algebra with a left H-coaction which is continuous in the weak sense. We define A C*-algebra C with the left G-coaction given below is called an induced coaction of (A, α) along φ if following three conditions hold.
is a well-defined * -homomorphism whose restriction to the unit ball is continuous from the strict topology of M(K(G) ⊗ A) to the strict topology of We can also define an induced coaction of a right coaction that is continuous in the weak sense along a homomorphism of locally compact quantum groups, by reversing left and right in the definition above. Remark 3.3. In the situation of Definition 3.2 above, an induced coaction C of (A, α) along φ is unique if it exists by the same argument from [32, Remark 6.8]. Indeed, for any C*-algebra D satisfying (i) and (ii), it holds that If D additionally satisfies (iii), we also have C = [DC] = D by symmetry. We will write Ind φ A for the unique induced coaction C of (A, α) along φ if it exists, and Ind φ α for its left G-coaction.
for the definition of Ind φ B. It follows from the previous Remark 3.3 that id ⊗f : As already mentioned, when φ : H → G is a homomorphism of regular locally compact quantum groups giving a closed quantum subgroup, then the construction of the induced coaction Ind φ A due to Vaes satisfies  (1) Consider the case of a trivial homomorphism 1 H→1 : H → 1 = G. Assume H is a locally compact group and X is a locally compact Hausdorff space with a proper left action of H.
(2) As a tautological example, for any homomorphism φ : to the unit ball is a homeomorphism onto its image with respect to the strict topologies. Indeed, the inverse is given by by the following Lemma 3.6, (iii) of Definition 3.2 follows from the next commutative diagram whose vertical arrows are homeomorphic on the unit balls with respect to the strict topologies.
Lemma 3.6. For a homomorphism φ : H → G of locally compact quantum groups, it holds . As for the converse inclusion, we observe id⊗∆ H is a right von Neumann algebraic H-coaction on M . Therefore by [31, Next, we are going to show the induction in stages Theorem 3.8, which gives a foundation of the remaining part. We begin with the following lemma.
Lemma 3.7. Let φ : H → G be a homomorphism of locally compact quantum groups, (A, α) be a C*algebra with a left H-coaction which is continuous in the weak sense, and H be a Hilbert space. If Ind φ A exists, then Ind φ (A ⊗ K(H)) also exists and is given by We consider the general case and for each finite dimensional subspace V ⊂ H we write 1 V for the unit of K(V).
Fix arbitrarily a net (x λ ) λ∈Λ and an element x in the unit ball of By replacing x and x λ with x * and x * λ , respectively, we also get Ad for any x ∈ Ind φ (A⊗ K(H)). Let Λ be the poset of all finite dimensional subspace of H with respect to the inclusions, and for each V ∈ Λ put . Later we will see the stronger compatibility of induced coactions with tensor products holds for the class of proper homomorphism in Corollary 4.4 and Proposition 6.3.

homomorphisms of locally compact quantum groups and (A, α) be a C*-algebra with a left H-coaction which is continuous in the weak sense. Suppose Ind φ A and
Proof. It follows from Lemma 3.6 that

Indeed, for any element x in the left hand side and any
In particular, this left F -coaction on C is continuous in the weak sense. Therefore it is enough to show that C satisfies (i) and (iii) for Definition 3.2 of Ind ψφ A. Since is also non-degenerate. For any z ∈ C we have Ad V ψ 12 (z) 13 13 , id ⊗α . Therefore C satisfies (i). We show (iii). By the strict continuity of Ad V ψ 12 (−) 13 : ) is a well-defined * -homomorphism and continuous on the unit ball from the strict topology of , and that this inclusion is continuous on the unit ball from the strict topology of We also have is well-defined and continuous on the unit ball from the strict topology of Therefore by composing this map with the strictly continuous map we obtain that the right hand side of (3 Quantum homogeneous spaces as quotients. Before going to the next section, we briefly discuss the coincidence of the two possibilities for quantum homogeneous spaces. In the following, for a C*-algebra A with a right coaction α of a locally compact quantum group G which is continuous in the weak sense, we write G and their strict topologies coincide on the unit balls. If A G exists, it is a good candidate for the fixed point algebra of A with the right coaction G (see (1) of Example 3.5). For a homomorphism φ : H → G of locally compact quantum group, we want to know when C r 0 (G) H and Ind φ C exist and coincide (see also the condition (d + ) in Remark 5.7). As another consequence of Lemma 3.7, we record a criterion for their coincidence.
and on the unit balls these strict topologies coincide by (iii). Now it is easy to see Ad V G 12 (−) 1 : is well-defined and continuous on the unit ball with respect to the strict topologies of M(K(G)) and M(K(G) ⊗ C r 0 (G) H ). We do not know this coincidence holds in general, but it is true in many tractable cases.
, and the strict topologies with respect to M(K(G)) and M(C r 0 (G/H)) coincide on the unit balls. Now it follows from the non-degenerate inclusion M(C r 0 (G/H)) ⊂ M(C r 0 (G)) H ⊂ L ∞ (G/H) = M(C r 0 (G/H)) that C r 0 (G/H) satisfies the definition of C r 0 (G) H . Especially this holds when G is discrete.
• This is also the case when H is a compact. This is essentially proved in [28,Theorem 5.1]. See also the remark after Theorem 4.7.

Induced coactions along a proper homomorphism
Next, we define proper homomorphisms of locally compact quantum groups, in a similar spirit as closed or open quantum subgroups.
Definition 4.1. Let G, H be locally compact quantum groups and φ : H → G be a homomorphism. We say φ is proper if φ r exists and φ r extends to a normal unital * -homomorphism L ∞ ( H) → L ∞ ( G). We still write φ r for this extension.
Let φ : H → G be a proper homomorphism of locally compact quantum groups in the sense of Definition 4.1 above. Then we can define a von Neumann bialgebra F by ( This is a well-defined locally compact quantum group by [ gives an open quantum subgroup, and im φ : Im φ → G is a homomorphism giving a closed quantum subgroup. This decomposition of φ is clearly unique (up to a natural isomorphism which makes the suitable diagram commute). Moreover if H is regular, then Im φ is also regular since its dual is regular by the last remark of Section 2.4.
We also note the result [13,

and put
, then the followings hold.
(1) A φ is a well-defined C*-algebra, and α φ : In the proof, we write P = p * p ∈ PZL ∞ ( H) = PZ(L ∞ ( H) ′ ). We recall from Section 2.4 that is a well-defined non-degenerate c.c.p. map. We also have as maps from by the injectivity and normality of φ r , we get 2) follows. By the second part of (4.2.2) we see A φ is actually a C*-algebra. It also follows that α φ := p 1 α(−)p 1 is a well-defined injective non-degenerate * -homomorphism of A φ → M(C r 0 (G) ⊗ A). Now it is easy to see (3) and (4).
We show (1). By composing p 2 (−)p * 2 after (4.2.1), it follows α φ : is a welldefined left G-coaction on A φ . This is continuous in the weak sense since by using (4.2.2) we get (i) and (ii) follow from (4.2.1) and (1). We show (iii). Since on Ind φ A we have where for the last inclusion we have used the fact by (3)   Proof. We decompose φ = (im φ)φ as in the remark after Definition 4.1. Note that Im φ is also regular. Then by Theorem 3.8, the statement is a direct consequence of [32, Theorem 7.2] for im φ and Lemma 4.2 forφ.
Proof. As for (1), by Theorem 3.8 and Lemma 4.2 we have and it follows Ind φ A = Ind im φ Aφ. Thus Ind φφ * Aφ = Ind im φ (φ * Aφ)φ = Ind im φ Aφ by (4) of Lemma 4.2. Next, we show (2). By (1) and regularity of V G , we have Here we note that when φ gives a closed quantum subgroup, Im φ is canonically identified with H itself and Aφ = A id H = A. By using (3.0.1), we see (3) holds since Now it is easy to see (4) by taking (B(L 2 ( G)) * ⊗id ⊗ id ⊗ id) Ad V G 12 (−) 134 to the following equality assured by (3)

4.2.
The compact case. We give a description of induced coactions along a homomorphism φ : K → G from a compact quantum group K = H to a locally compact quantum group G by using cotensor products. Although the content of this section might be well-known to the experts, we contain it since we could not find appropriate references. First, we fix some notation about compact quantum groups. For details, see [22] for example. Then we introduce cotensor products of C*-algebras by following [24], [10], [36]. Especially, we closely follow the notation from Section 2 of [36]. Consider a compact quantum group K with its faithful Haar state h on C r 0 (K). A pair ρ = (H ρ , u ρ ) of a finite dimensional Hilbert space H ρ and a unitary matrix u ρ ∈ U(K(H ρ ) ⊗ C r 0 (K)) is called a finite dimensional unitary (right) representation of K if when we express u ρ = (u ρ i,j ) i,j with respect to some orthonormal basis of H π , it holds ∆ K (u ρ i,j ) = dim H ρ k=1 u ρ i,k u ρ k,j for all i, j. We write Rep(K) for the C*-tensor category of finite dimensional unitary representations, Irr(K) ⊂ Rep(K) for the set of representatives of isomorphism classes of irreducible objects, and 1l = 1l K ∈ Irr(K) for the trivial 1-dimensional representation of K. Any ρ ∈ Rep(K) can be written as a direct sum of elements in Irr(K), and for π ∈ Irr(K) we write π ≤ ρ if ρ contains π as a direct summand. For each π ∈ Irr(K), we define the positive definite matrix F π = (F π i,j ) i,j ∈ B(H π ) as the unique invertible intertwiner from the representation π to its double contragredient representation π cc (which need not be unitary) with trF π = tr((F π ) −1 ). Then the quantum dimension of π is denoted by dim q π := trF π . As for F π , all we need here is the following orthogonality relation. For all π, ̟ ∈ Irr(K), 1 ≤ i, j ≤ dim H π , and 1 ≤ k, l ≤ dim H ̟ , it holds h(u π i,j u ̟ k,l * ) = (dim q π) −1 δ π,̟ δ i,k F π l,j . Hereafter for each π ∈ Irr(K), we choose an orthonormal basis (e π i ) i of H π such that it diagonalizes (F π i,j ) i,j . We define O(K) := It is a consequence of Peter-Weyl theorem that O(K) is a norm-dense Hopf * -subalgebra of C r 0 (K) and (u π i,j | π ∈ Irr(K), 1 ≤ i, j ≤ dim H π ) forms its linear basis. Fix a right K-C*-algebra (A, α). Its spectral subspace of each ρ ∈ Rep(K) is denoted by A ρ := {a ∈ A | α(a) ∈ A ⊙ O(K) ρ }. For a ∈ A ρ , we take the elements a π i,j ∈ A for π ∈ Irr(K) with π ≤ ρ and a π i,j ⊗ u π j,i . We observe a π i,k ⊗ u π k,j ⊗ u π j,i .
Then we see that α(a π i,j ) = dim H π k=1 a π i,k ⊗ u π k,j , and a = a π i,i , by the linear independence of {u π i,j } π,i,j and the injectivity of α. Especially a π i,j ∈ A π . It holds that (id A ⊗h)α : A ։ A 1 l is a conditional expectation onto the non-degenerate C*-subalgebra A 1 l ⊂ A. More generally for any ρ ∈ Rep(G), a calculation using a π i,j shows that the continuous A 1 lbimodule map defined by satisfies p A ρ ↾ Aρ = id Aρ and p A ρ (A) = A ρ . It turns out that the * -subalgebra Now we define the cotensor product by A for each ρ ∈ Rep(K). By a similar calculation as that for p A ρ , we can show the following lemma. Then Proof. It is clear that where A ⊗ 1 * B is the right K-C*-algebra (A ⊗ B, σ 12 (id ⊗α)σ). For all π, ̟ ∈ Irr(G) with either π = ̟ or π ≤ ρ, we see r ρ (A π ⊗ ̟ B) = 0. For any a ∈ A ρ and b ∈ ρ B, by using a π i,j and b π i,j we calculate Thus we get r ρ (A ρ ⊙ ρ B) ⊂ A  Proof. First, we note A 1 l ⊗ 1 l B ⊂ A ⊗ B is a non-degenerate C*-subalgebra. We can check For each ρ ∈ Rep(K) it holds (γ ⊗ id)r ρ = (id ⊗r ρ )(γ ⊗ id) on A ⊗ B by definition, and we see Thus it follows γ ⊗ id : ) is a well-defined non-degenerate * -homomorphism which gives a continuous left G-coaction.
Theorem 4.7. Consider a homomorphism φ : K → G of locally compact quantum groups and assume K is compact. Let (A, α) be a left K-C*-algebra. Then Ind φ A exists and is given by Especially, by letting A = C we get the last case of Example 3.10. Note that we do not need regularity on G. We also note that a homomorphism φ : K → G of locally compact quantum groups is always proper when K is compact, because C r 0 ( K) is a direct sum of finite dimensional C*-algebras.
Thus for any c ∈ K(G) ⊗ (C r 0 (G) 1 l ) and a ∈ 1 l A, it holds that where we have equipped K(G) with a trivial coaction, and used r K(G)⊗C r 0 (G), A ρ = id ⊗r C r 0 (G), A ρ . It follows from the strict continuity of Ad V G 12 (−) 13 :

Analogue of base change
We go back to the case of induced coactions along general proper homomorphisms. From now we restrict our attention to regular locally compact quantum groups to assure the existence of induced coactions. Throughout this section, we consider the following situation, which changes quantum groups coacting on C*-algebras.
Condition 5.1. Consider the following diagram of regular locally compact quantum groups and homomorphisms, which commutes in the sense of φ = ıθ, such that there is a * -isomorphism δ : C r 0 (G) ∼ − → Ind θ  * C r 0 (H) which makes the following diagram commutes, Here we remark that by using the fact [C r

it can be seen that the diagram (5.1.2) commutes if and only if (id ⊗δ)∆
Similarly, this is also equivalent to (δ ⊗ id)∆ G = (id ⊗φ * ∆ G )ı * ∆ G as maps from C r 0 (G). We want to compare Ind φ A and Ind θ  * A for a left H-C*-algebra A. First, we are going to construct the map to compare two induced coactions formally. It immediately turns out that it is isomorphic in some special case. (
We have a non-degenerate inclusion is a non-degenerate C*-subalgebra, and ∆ E ⊗ id ⊗ id gives well-defined continuous E-coaction on it. Therefore the argument of Remark 3.3 shows Next, we show (2). By (1) it is enough to show δ(Ind φ C) (Ind θ C) ⊗ 1 K(H) = δ(Ind φ C). First, we observe that the restriction of δ gives a * -isomorphism δ : M(C r 0 (G)) H ∼ − → M(Ind θ C r 0 (H)) ∩ M(C r 0 (E)) F ⊗ 1 K(H) . Then by Remark 3.4 for C → M(C r 0 (H)) we get an injective non-degenerate * -homomorphism By construction f is a left E- * -homomorphism. Thus by the normality of ı r and (iii) of Definition 3.2 for Ind φ C we obtain As for the converse inclusion, let (e λ ) λ be an approximate unit of Ind θ C and ω be a normal state on B(L 2 (E)). Then we have (ωı r ⊗ id)∆ G f (e λ ) λ − → 1 strictly in M(C r 0 (G)), and thus the two inclusions in (5.2.1) are equalities. By taking δ the desired claim follows.
Next, we suppose φ and θ are proper, and state the analogue of base change formula.  (c) There is some non-degenerate C*-subalgebra D ⊂ M(C r 0 (G)) which is a left G-C*-algebra by

Indeed, clearly (b) implies (c). (c) implies (d) because by (iii) of Definition 3.2 for Ind
. It is proved at the beginning of the proof of (2) in Lemma 5.2 that (d) implies (b).
To prove Theorem 5.3, we decompose the homomorphisms as we did in Theorem 4.3, and reduce the statement to the cases of closed quantum subgroups and duals of open quantum subgroups. In (4) of Lemma 5.5 next, we prove the latter case at the same time with the reduction step. (1) There is a unique homomorphism : Im φ → Im θ such that the two squares in the following diagram are commutative, The left square of (5.5.1) satisfies Condition 5.1. The well-defined left Im θ- * -isomorphism Proof. (1) follows from the fact that C u 0 (Im φ) can be canonically identified with the image of φ u : C u 0 (G) → C u 0 (H), and so is C u 0 (Im θ).
We show (2). The commutativity of the diagram (5.1.2) for this square is easy. By using (2) of Example 3.5 and Corollary 4.4 we calculate Thus again a similar calculation shows From this and Lemma 4.2 we obtain Therefore the desired * -isomorphism follows from the injectivity ofθ r ⊗ id. Now (b) of Theorem 5.3 for this square holds because of (1) in Lemma 5.2 and the fact Indφ C ∼ = C ∼ = Indθ C. As for (3),δ is well-defined as a map of C r By definition,δ makes the diagram (5.1.2) commute. By using Corollary 4.4 and (5.5.2), we get is a well-defined * -isomorphism because δ is a * -isomorphism by the assumption. (b) of Theorem 5.3 for the original square (5.1.1) implies (b) for the right square of (5.5.1) because Ind im θ C = Ind θ C and Ind im φ C = Ind φ C hold by Corollary 4.4.
We proceed to prove Theorem 5.3. The remaining case is the case when φ and θ give closed quantum subgroups. In order to prove this, we will concretely realize a left F -C*-algebra B with Ind θ B ∼ = ı * Ind φ A by following the proof of [32,Theorem 9.2], and reduce the conclusion to  * α(A) ∼ = B.
Proof of Theorem 5.3. We only have to prove (b) implies (a). By (3) and (4) of Lemma 5.5, it suffices to consider the case when φ and θ give closed quantum subgroups. Consider the C*-algebras which are well-defined right E-C*-algebras with the coactions given by the restrictions of Ad V E 14 (−) 123 . The restrictions of Ad V E 12 (−) 134 give right E- * -isomorphisms from C φ and C θ to E ⋉ r (δ ⊗ id) Ind φ A and E ⋉ r Ind θ  * (αA), respectively.
Let us assume (b). Then we can consider a non-degenerate * -homomorphism where the last map is well-defined and non-degenerate because is a non-degenerate C*-subalgebra, and thus I θ It turns out I θ gives a welldefined continuous right E-coaction on I θ . Therefore Ad V F 14 (−) 123 is a well-defined continuous right F -coaction on I θ 1 * C φ I θ 1 by the argument from [32, Theorem 9.2] (see Lemma 5.6 below). Since we can use (the proof of) [32,Theorem 6.7]. As a consequence, it holds that ) by the definition of δ, we obtain from the arguments so far, On the other hand, we also have Therefore if it holds , and by taking (K(E) * ⊗ id K(E)⊗K(H)⊗A ) Ad V E 12 (−) 134 it follows (δ ⊗ id) Ind φ A = Ind θ  * (αA), which will complete the proof. But we calculate by writing and thus (5.5.3) holds as desired.
In the proof above, we have used an argument taken from [32,Theorem 9.2]. We record this for the convenience of the reader.

Lemma 5.6. Let φ : H → G be a homomorphism of regular locally compact quantum groups which gives a closed quantum subgroup, H be a Hilbert space, and A ⊂ B(H) be a non-degenerate C*-subalgebra. Let α : A → B(H)⊗L ∞ ( H) be a * -homomorphism such that (id⊗ φ r )α is a continuous right G-coaction on A. Then α is a continuous right H-coaction on A.
Proof. Clearly α is injective on A. By putting Thus α : A → M(A ⊗ K(H)) is a well-defined non-degenerate * -homomorphism. By the injectivity of φ r it holds (α ⊗ id K(H) )α = (id B(H)⊗ ∆ H )α. Then it follows from the regularity of H that , which shows the desired statement. To use Theorem 5.3, the condition (b) looks still technical. But in many practical cases, this follows automatically from other assumptions, as we list here.
• As observed by [32] and [23], (c) of Remark 5.4 holds when the square (5.1.1) is given by (generalized) quantum doubles. See Example 5.9 below for detail. This is our motivating example of Theorem 5.3. (d + ) Ind φ C also gives the induced coaction C r 0 (G) H of the right H-C*-algebra C r 0 (G) * φ along the homomorphism 1 H→1 . To see this, we observe as in the beginning of the proof of (2) in Lemma 5.2 that (Ind θ C) ⊗ 1 K(H) ⊂ δM(C r 0 (G)) H = δM(C r 0 (G) H ). This inclusion is non-degenerate by (iii) of Definition 3.2 for C r 0 (G) H , which shows (d) of Remark 5.4. As we saw in Example 3.10, (d + ) holds when G is a locally compact group, φ gives an open quantum subgroup, or H is a compact quantum group.
We give some examples. From now, we use the notation and terminologies from Appendix A except Proposition A.7 and Example A.8, which concern twisted tensor products.
Example 5.8. Let F, G, H be regular locally compact quantum groups, m ∈ Aut(C r 0 (G) ⊗ C r 0 (F )) extend to a matching on G and F , and n ∈ Aut(C r 0 (H) ⊗ C r 0 (F )) extend to a matching on H and F . Consider a proper homomorphism φ : H → G satisfying (A.3.1). Then we have a homomorphism φ op ⊲⊳ id F : D(n) → D(m) by Proposition A.3 and the commutative square Example 5.9. Consider a homomorphism φ : H → G of strongly regular locally compact quantum groups that gives a closed quantum subgroup and a H-Yetter-Drinfeld C*-algebra (A, α, α). It is shown by [23,Proposition 3.4] that Ind φ ((id ⊲⊳ φ) * A) has a canonical structure of an G-Yetter-Drinfeld C*-algebra. Note that with Remark 3.1 taken into account, this statement only requires regularity.
Example 5.10. Consider a sequence of regular locally compact quantum groups with K being compact, which is exact in the sense of ι, φ giving closed quantum subgroup and C r 0 (H/K) = φ r C r 0 (G) (see (3) of Example A.5). Then satisfies Condition 5.1, and condition (d) of Theorem 5.3 by Remark 5.7. Therefore for any left H-C*algebra A, it holds A φ = Ind K→1 ι * A = 1 lK ι * A.

Morita and K-theoretic properties
In this section, we consider the functors between several kinds of the categories of coactions which are induced by induced coactions along a proper homomorphism φ. We prove several properties of such functors which are known when φ gives a closed quantum subgroup in [23] and [32]. We say left G-(A, B)correspondences (E, π) and (F , ̟), are left G-unitarily isomorphic if there is a unitary U ∈ UL B (E, F ) satisfying (Ad U )π = ̟, such that Ad We will also use the terminologies from Appendix A, especially those about twisted tensor products.
Definition 6.1. Let G be a regular locally compact quantum group.
(1) Let C*alg G be the category of left G-C*-algebras as objects, and G- * -homomorphisms A → B as morphisms from A to B for A, B ∈ C*alg G . (2) Let Corr G be the category of left G-C*-algebras as objects, and morphisms as follows. For A, B ∈ Corr G , a morphism from A to B is a left G-unitary isomorphism class of left G-(A, B)correspondences (E, π) with non-degenerate π. The composition of morphisms is given by inner tensor products. (3) Suppose C r 0 (G) is separable. Let KK G be the category of separable left G-C*-algebras as objects, and the G-equivariant bivariant K-group KK G (A, B) as the set of morphisms from A to B for A, B ∈ KK G . The composition of morphisms is given by Kasparov products. We refer [2] for details.
We will use the following facts from [23] about KK G for a regular locally compact quantum group G with C r 0 (G) separable. Let F : C*alg G → A be a functor to an additive category A with following three properties.
• For A, B ∈ C*alg G and a left G- * -homomorphism f : where f t (a) denotes (f (a))(t) for all a ∈ A and t ∈ [0, 1].
• For a split exact sequence of left G- * -homomorphisms 0 Then there is an additive functorF : KK G → A such thatF ι ∼ = F , where we write ι : C*alg G → KK G for the canonical functor. Moreover, suchF is unique up to a natural isomorphism and ι itself satisfies the three properties above. Also, there is a canonical structure of a triangulated category on KK G . Remark 6.2. Consider a homomorphism φ : H → G of regular locally compact quantum groups, and C ∈ {C*alg, Corr, KK}, where we additionally assume separability of C r 0 (G) and C r 0 (H) when C = KK. (1) We still write φ * : C G → C H for the functor given by restrictions of left coactions along φ.
(2) When φ is proper, induced coactions canonically induces the well-defined functor φ * : C*alg H → C*alg G . When φ gives a closed quantum subgroup, this is shown in the last paragraph of Section 2 in [23]. When φ gives an open quantum subgroup, this follows from f (Aφ) ⊂ Bφ for any left H- * -homomorphism f : A → B of left H-C*-algebras. The general case holds by Theorem 3.8. We can also check that this functor preserves split exact sequences. (3) Suppose C = Corr or KK. We also write φ * : C H → C G for the well-defined functor given by . This construction is compatible with compositions of homomorphisms by Baaj-Skandalis duality.
As for (3), we note that when φ does not give a closed quantum subgroup induced coactions along φ need not induce a functor C H → C G in a direct way (see the remark before Proposition 6.4).
6.1. Induced coactions and twisted tensor products. We want to regard the functors φ * : C H → C G of (3) in Remark 6.2 as the functors induced by induced coactions. This is justified because it holds φ * (A) ∼ = Ind φ ( G op ⋉ r G ⋉ r A) as a special case of Proposition 6.3 next by (2) of Example A.8. When φ gives a closed quantum subgroup, this is proved in [23,Theorem 3.6]. Their proof can be easily adapted to our situation, but nevertheless we include the proof since their statement is made in a slightly different way. The crucial point for the proof is an argument from [32, Theorem 8.2] that for any left G-C*-algebra (B, β), it holds We again note that the argument is still valid when G and H are regular.
Proof. We put p := p φ associated with φ . Therefore by Corollary 4.4, we have Here for the last three equalities we have used (6.3.1), and p 1 W ψφ = p 1 (φ r ⊗ id)W ψ im φ = W ψ im φ p 1 .
It is easy to check that this equality preserves left G-and E-coactions.
6.2. Imprimitivity. For a proper homomorphism φ : H → G of regular locally compact quantum groups, sometimes we want to know when the functor φ * becomes genuine Ind φ , for the purpose of calculation. For left G-C*-algebras (A, α), (B, β) and a left G-(A, B)-correspondence (E, π) with non-degenerate π, the canonical non-degenerate left H- * -homomorphism ι : and this is a well-defined right (Ind φ B)-Hilbert module with a canonical continuous left G-coaction (which is different from an induced corepresentation in nature). By definition we also have a nondegenerate left G- * -homomorphism µ : is a well-defined left G-(A, B)-correspondence with non-degenerate Ind φ π := µ(id ⊗π).
When φ gives a closed quantum subgroup, the imprimitivity theorem for reduced crossed products [32,Theorem 7.3] assures φ * A is naturally isomorphic to Ind φ A for any A ∈ Corr H . For morphisms (E, π) : a left G-unitary equivalence, by the construction of Ind φ using I φ .
For a general proper φ, this is no longer true since Ind φ need not preserve equivariant Morita equivalence. For example, consider φ := 1 K→1 : K → 1 for a compact quantum group K. Then the left K-C*-algebras C and (K(K), Ad W K 12 * (−) 2 ) are left K-Morita equivalent, but Ind φ C = C and are not Morita equivalent unless K is trivial. To get the well-behavior of Ind φ , we consider a relative variant of freeness as follows.
Then Ind φ gives a well-defined functor Corr H φ → Corr G which is naturally isomorphic to φ * ↾ Corr H φ .
We note in the situation of the proposition above, (A, α) ∈ Corr H is an object of Corr H φ if and only if it is an object of Corr H φ forφ : H → Im φ, which is equivalent to say α(A)(φ r (C r 0 (Im φ)) ⊗ A) = C r 0 (H) ⊗ A. We also note that for a regular locally compact quantum group F and a left F -C*-algebra A ∈ Corr H φ for any left H-C*-algebra A. For a right Hilbert A-module E and an (A, B)-correspondence (F , ̟), we write E ⊙ ̟ F for the tensor product of bimodule in a purely algebraic sense.
Proof. As we saw, the result holds when φ gives a closed quantum subgroup. By decomposing φ = (im φ)φ and using Theorem 3.8, it suffices to consider the case when φ gives an open quantum subgroup.
Let A, B ∈ Corr H . Take a left H-Hilbert A-module (E, κ E ) and a morphism (F , ̟) : Indeed, from this it will follow that Ind φ defines a well-defined functor Corr H φ → Corr G , which must be naturally isomorphic to Thus when we put p := p φ , it follows Therefore when we write κ for the left H-coaction of E ⊗ ̟ F , we obtain  6.3. Adjointness. As the last topic of this section, we record the adjointness of induced coactions and restriction. For a while, we fix a proper homomorphism φ : H → G of regular locally compact quantum groups which has a cocompact image in the sense of Ind φ C being unital. Suppose φ r exists, and let C ∈ {C*alg, Corr, KK}, where we additionally assume separability of C r 0 (G) and C r 0 (H) when C = KK. When φ gives a cocompact closed quantum subgroup and G is coamenable, it is proved in [23,Proposition 4.7] that there is an adjunction φ * ⊣ φ * as functors between KK G and KK H , as the quantum version of the result for locally compact groups in [19]. We note that their proof is quite analogous to Frobenius reciprocity for unitary representations of a compact group and its closed subgroup (see [11] for example). By slightly modifying their construction of the counit denoted by κ below, we can loosen the assumption on the coamenability of G into the existence of φ r as follows.
First, let C = C*alg. Consider a proper homomorphism φ : H → G with the conditions above. For all (A, α) ∈ C*alg G and (B, β) ∈ C*alg H we have by Proposition 6.3, and give the natural transformations η : id C*alg G → φ * φ * and κ : φ * φ * → id C*alg H , respectively. For all (A, α) ∈ C*alg G and (B, β) ∈ C*alg H , it holds and Therefore η and κ are the unit and counit which assure φ * ⊣ φ * as functors between C*alg G and C*alg H . Now suppose that C = Corr or KK, and that φ gives a closed quantum subgroup. Then Proposition 6.4 and the universality of KK H show φ * ⊣ φ * as functors between C G and C H , as claimed above.
Next, we seek such adjointness when φ gives an open quantum subgroups and C = Corr or KK. When C = KK, it is proved as a special case of [37,Theorem 4.8] that φ * ⊣ φ * as functors between KK G and KK H . We remark that the proof still works for C = Corr. Thus by Baaj-Skandalis duality it follows φ * ⊣ φ * as functors between C G and C H , where C = Corr or KK.
By combining the arguments so far, we obtain the following.
Proposition 6.6. Let φ : H → G be a proper homomorphism of regular locally compact quantum groups with Ind φ C being unital, and C ∈ {C*alg, Corr, KK}, where we additionally assume separability of C r 0 (G) and C r 0 (H) when C = KK. Suppose φ r exists. Then φ * ⊣ φ * as functors between C G and C H . As already mentioned, this adjointness in the case of a closed quantum subgroup can be considered as an analogue of Frobenius reciprocity. On the other hand, let φ := 1 K→1 : K → 1 for a compact quantum group K with separable C r 0 (K), as a special case when φ gives an open quantum subgroup. Then ) holds for any A ∈ KK and B ∈ KK K , which is exactly (the quantum version of) Green-Julg theorem, [35,Théorème 5.10].
Consider a homomorphism φ : H → G of regular locally compact quantum groups which gives a closed quantum subgroup. In [23], they considered induced coactions along φ to give a functor C D(H) → C D(G) . In our notation, this is given by (φ ⊲⊳ id) * (id ⊲⊳ φ) * : C D(H) → C D(φ) → C D(G) . As an immediate consequence from the argument of Proposition 6.6, we have the following adjointability. Proof. Note that φ ⊲⊳ id and the dual of id ⊲⊳ φ give open quantum subgroups by Example A.5. Thus the statement follows from the fact that (φ ⊲⊳ id) * ⊣ (φ ⊲⊳ id) * and (id ⊲⊳ φ) * ⊣ (id ⊲⊳ φ) * .

7.
Example: double crossed products with a discrete abelian group In this section, we stick to the following situation.
Condition 7.1. Let F, G, H be regular locally compact quantum groups and Γ be a discrete abelian group. Consider the following sequences of homomorphisms of locally compact quantum groups.
Moreover, we assume they are "split exact" (with common π) in the sense of satisfying the three conditions as follows.
Remark 7.2. Assume the situation of Condition 7.1. By (3) of Example A.5 we have φ r C r 0 (G) = Ind Γ→1 ı * C r 0 (H), and similarly ψ r C r 0 (F ) = Ind Γ→1  * C r 0 (H). By using this and the commutativity of C 0 (Γ), it follows for any x ∈ (φ r C r 0 (G)) ⊗ C 0 (Γ), Therefore Ad W π gives a * -automorphism on (φ r C r 0 (G)) ⊗ C 0 (Γ), and induces a matching m on G op and Γ. By the same reason we also have a matching n on G op and Γ coming from a * -automorphism Ad W π on (ψ r C r 0 (F ))⊗C 0 (Γ). By (3) of Example A.5 we see φ ⊲⊳ id and ψ ⊲⊳ id are well-defined homomorphisms which give open quantum subgroups.
When F, G, H are compact, such a situation arises from a more general treatment in [6]. Here we give an example coming from finite groups.
The left D(n)-coaction of T ′ n (A) is canonically determined by the left D(π)-coaction of (ψ ⊲⊳ id) * T ′ n (A) as a subset of Γ ⋉ r A. Thus we obtain We have a left H-and Γ- * -homomorphism where s, t ∈ Γ, a ∈ t ( * A), and ǫ Γ is the counit of C * (Γ). This map is actually a * -isomorphism, whose inverse can be given by where we have used the fact that for any s, t ∈ Γ it holds By combining (7.4.2) with the left D(m)- * -isomorphism α : Next, we show the case of C = Corr. We observe  * (Γ ⋉ r A) ∈ Corr Γ Γ→1 for any π-Yetter-Drinfeld C*-algebra (A, α, α), since Thus Example 5.10 and Remark 6.5 imply T ′ n is a well-defined functor with T n ∼ = T ′ n . Now it is enough to show id Corr D(m) ∼ = T ′ m (ψ ⊲⊳ id) * T ′ n (φ ⊲⊳ id) * . But for any morphism (F , ̟) : A → B in Corr D(m) , we can describe T n (φ ⊲⊳ id) * (F ) as a corner of (7.4.1) for K B (F ⊕ B). By using this we can check that η constructed in the case of C = C*alg still gives a natural isomorphism ( T n (φ ⊲⊳ id) * (−)) φ⊲⊳id ∼ − → id Corr D(m) . Finally, we consider the case of C = KK. Note that the separability of C r 0 (H) implies separability of C r 0 (G), C r 0 (F ), C 0 (Γ) by Condition 7.1. As a functor of C*alg D(π) → C*alg D(n) , T ′ n preserves split exact sequences and tensor products with the closed interval [0, 1]. We also know from the previous case of C = Corr that T ′ n preserves equivariant imprimitivity bimodules. Therefore from the universal property of KK D(π) it follows T ′ n : KK D(π) → KK D(n) is well-defined. It also follows from Baaj-Skandalis duality and η in the case of C = C*alg that T ′ n ∼ = T n and ( T n (φ ⊲⊳ id)(−)) φ⊲⊳id ∼ = id KK D(m) .  need not be isomorphic, even in the finite dimensional case. In this sense, the corollary can be interpreted as a kind of no-go theorem for reconstructing quantum groups using Corr and KK. We remark this categorical equivalence need not send C ∈ C D(m)  Theorem 7.4. But by the construction of T ′ m , it is easy to see that for any A ∈ C D(π) whose coaction is trivial, T ′ m (A) ∈ C D(m) also has a trivial coaction and canonically * -isomorphic to A.
Finally, we briefly discuss the implications for the quantum analogue of torsion phenomena and Baum-Connes conjecture. Let Λ be a discrete quantum group. In the context of the quantum analogue of Baum-Connes conjecture, finite dimensional left Λ-C*-algebras up to left Λ-Morita equivalence are considered to reflect the torsion phenomena of Λ (or Λ op ). In [18], torsion-freeness for Λ is defined to state the quantum analogue of Baum-Connes conjecture for such a discrete quantum group. Recently a formulation of the quantum analogue of Baum-Connes conjecture for Λ which need not be torsion-free is proposed in [1]. Both works use certain decompositions of the triangulated category KK Λ . Now consider the situation of Condition 7.1 with H being compact and Γ being finite, and we have C D(m) ≃ C D(n) as in Theorem 7.4. It is easy to see Corr D(m) ≃ Corr D(n) preserve finite dimensionality of equivariant C*-algebras, and there is a bijective correspondence between the equivariant Morita equivalence classes of finite dimensional left D(m)-C*-algebras, and those for D(n), which preserves finite direct sums. Also when C r 0 (H) is separable, we can see the triangulated equivalence KK D(m) ≃ KK D(n) preserves the decompositions of the two triangulated categories given by [1, Corollary 5.1].

Appendix A. Constructions for locally compact quantum groups and coactions
In this section we gather some constructions which are not directly related to the general theory of induced coactions. For a C*-algebra A, we write Aut(A) : A.1. Double crossed products and homomorphisms. We collect some definitions and properties about double crossed products of locally compact quantum groups by following [5]. . Example A.2. Let φ : H → G be a homomorphism of locally compact quantum groups. Then m := Ad W φ ∈ Aut(C r 0 (H)⊗C r 0 ( G)) gives a matching of H op and G. Then we have a locally compact quantum group D(Ad W φ ) called the generalized quantum double, for which we simply write D(φ). When φ = id G we also write D(G) = D(id G ) for short, which is called the quantum double of G. If G, H are regular, then C r 0 (D(φ)) = C r 0 (H) ⊗ C r 0 ( G). See Section 8 of [5] for detail. Before stating the next proposition, we note some facts needed from [5]. (2) If φ has (P), then φ op ⊲⊳ id also has (P).
We consider (3). We assume φ is proper and we show Φ is proper. Since V φ = ( φ r ′ ⊗ id)V H , the comparison of (ω ⊗ id ⊗ id)W D(n) and (ω ⊗ id ⊗ id)W Φ for each ω ∈ B(L 2 (D(n))) * shows it is enough to show that Ad(Z m * Z n ) : Ad Z n * L ∞ ( F ) 2 → Ad Z m * L ∞ ( F ) 2 and φ r ′ : L ∞ ( H) ′ → L ∞ ( G) ′ extend to some well-defined normal * -homomorphism Indeed, this map will give Φ r . Then if φ r is surjective, Φ r will be also surjective. When we write G for the bicrossed product associated with m, we have for all x ∈ L ∞ ( G) ′ , and for all y ∈ L ∞ ( F ), J m Z m (Ad Z m * y 2 ) * Z m * J m = U m R F (y) 2 U m * ∈ Ad U m L ∞ ( F ) 2 .
Similar equalities hold for n. By [16,Proposition 5.45], it is enough to show that φ r and id B(L 2 (F )) extend to some well-defined normal * -homomorphism H ⋉ Ad U n B(L 2 (F )) → G ⋉ Ad U m B(L 2 (F )) via the (φ op ⊲⊳ id F ) r C r 0 (D(m)). Then by the similar reasoning as above shows C r 0 (D(n)/K op ) = (φ op ⊲⊳ id F ) r C r 0 (D(m)). Especially, it follows φ op ⊲⊳ id F gives an open quantum subgroup. A.2. Yetter-Drinfeld C*-algebras and twisted tensor products. In this section, we collect some definitions and properties about Yetter-Drinfeld C*-algebras, as another picture for left coactions of a generalized quantum double, and their twisted tensor products by following [23], [27], [21].
Definition A.6. Let φ : H → G be a homomorphism of locally compact quantum groups.
(1) Let A be a C*-algebra, α be a continuous left H-coaction on A and α be a continuous left G-coaction on A. Then A = (A, α, α) is a (left) φ-Yetter-Drinfeld C*-algebra if the following diagram commutes, .  We will make use of the notion of twisted tensor products in Section 6 and Section 7. We list its fundamental properties as follows. All of them are contained in [21] and [23], or can be shown easily by using the techniques in them. As for the notation of the categories in (3), see Definition 6.1 and the remark after that.
(1) A ⊠   (1) We write C φ to be C endowed with the trivial structure of φ-Yetter-Drinfeld C*-algebra. Then (2) Consider the 1 G op →G -Yetter-Drinfeld C*-algebra (C r 0 ( G), ∆ cop G , ∆ G ). Then is a well-defined left G op - * -isomorphism, if we regard the right G-coaction of G ⋉ r A as a left G op -coaction. See Section 6.3 in [21] for detail. When G is regular, the Baaj-Skandalis duality [3] says naturally. Here we have identified