Quantization of subgroups of the affine group

Consider a locally compact group $G=Q\ltimes V$ such that $V$ is abelian and the action of $Q$ on the dual abelian group $\hat V$ has a free orbit of full measure. We show that such a group $G$ can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of $G$ on $L^2(V)$; (2) by twisting the coproduct on the group von Neumann algebra of $G$ by a dual $2$-cocycle obtained from the $G$-equivariant Kohn-Nirenberg quantization of $V\times\hat V$; (3) by considering the bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$ both isomorphic to $Q$. In the simplest case of the $ax+b$ group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum $ax+b$ group of Baaj-Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups $G$. Using recent results of De Commer we show that a class of $G$-Galois objects is parametrized by certain cohomology classes in $H^2(G;\mathbb T)$. This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in $H^2(G;\mathbb T)$ can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group $G$ with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual $2$-cocycles such that the corresponding quantization of $G$ is neither commutative nor cocommutative.

Consider a locally compact group G = Q V such that V is abelian and the action of Q on the dual abelian group V has a free orbit of full measure. We show that such a group G can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of G on L 2 (V ); (2) by twisting the coproduct on the group von Neumann algebra of G by a dual 2-cocycle obtained from the Gequivariant Kohn-Nirenberg quantization of V ×V ; (3) by considering the bicrossed product defined by a matched pair of subgroups of Q V both isomorphic to Q.
In the simplest case of the ax + b group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum ax + b group of Baaj-Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups G. Using recent results of De Commer we show that a class of G-Galois objects is parametrized by certain cohomology classes in H 2 (G; T ). This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in H 2 (G; T ) can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group G with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual 2-cocycles such that the corresponding quantization of G is neither commutative nor cocommutative.
It can be explicitly quantized using the Jordanian twist found independently by Coll-Gerstenhaber-Giaquinto [8] and Ogievetsky [27]. 1 This twist and its generalizations have been extensively studied, see, e.g., [20], [21]. It is particularly popular in physics literature, as it can be used to construct the κ-Minkowski space [6], which by [22] can also be obtained from a bicrossed product construction.
If we want to make sense of Ω as a unitary operator on L 2 (G ×G), since the elements x and y are skew-adjoint, our best bet is to take h ∈ iR, but then we still have a problem with the logarithm, as the spectrum of y is the entire line iR. A correct analytic analogue of the Jordanian twist was found by Stachura [30], see formula (3.31) below, but it turns out that, similarly to the case of SU (1,1), it is important to work with the entire nonconnected ax + b group. What to do in the connected case remains an open problem.
In fact, in an earlier paper [4] the first two authors found a universal deformation formula for the actions of the connected component of the ax + b group (and, more generally, of Kählerian Lie groups) on C * -algebras. Unfortunately, despite the claims in [26] and [5], this formula turned out to define a coisometric but nonunitary dual 2cocycle, which is the term we prefer to use in the analytical setting instead of the "twist". See Remark 2.12 below and erratum to [5] for further discussion.
If we do consider the nonconnected ax + b group, then even earlier, Baaj and Skandalis constructed its quantization as a bicrossed product of two copies of R * [29]. One disadvantage of this construction is that from the outset it is not clear how justifiable it is to call their quantum group a quantization of the ax + b group. Some justification was given later by Vaes and Vainerman [33].
The present work grew out of the natural question how the constructions in [29], [4] and [30] are related. As we already said, we found out that [4] does not lead to a unitary cocycle and therefore cannot actually be used to quantize the ax + b group. But the constructions in [29] and [30] turned out to be equivalent, as was conjectured by Stachura. Furthermore, we found an interpretation of the Jordanian twist/Stachura cocycle in terms of the Kohn-Nirenberg quantization, which allowed us to construct quantum analogues of a class of semidirect products Q V . We also realized that these constructions have a very natural description within De Commer's analytic version of the Hopf-Galois theory [11,12].
In more detail, the main results and organization of the paper are as follows. After a short preliminary section, we begin by discussing G-Galois objects for general locally compact groups G in Section 2. These are von Neumann algebras equipped with actions of G that are in an appropriate sense free and transitive. For compact groups such actions are known in the operator algebra literature as full multiplicity ergodic actions.
Using recent results of De Commer [12] we show that the G-Galois objects with underlying algebras factors of type I are classified by certain second cohomology classes on G (Theorem 2.4). For finite groups such a result quickly leads to a complete classification of G-Galois objects obtained by Wassermann [34] (although the result is not very explicit there, see [25]) and Davydov [10]. For infinite groups the situation is of course more complicated, as in general there exist Galois objects built on non-type-I algebras.
We show next that under extra assumptions a G-Galois object of the form (B(H), Ad π), where π is a projective representation of G on H, defines a dual unitary 2-cocycle (Proposition 2.9). We do not know whether these assumptions are always automatically satisfied, but we show that they are if π is a genuine representation (Theorem 2.13). This implies that if the group von Neumann algebra W * (G) of a nontrivial group G is a type I factor, then there exists a canonical nontrivial cohomology class of dual cocycles on G. This gives probably the shortest explanation why a quantization of, for example, the ax +b group exists at the operator algebraic level. At the Lie (bi)algebra level this is related to Drinfeld's result on quantization of Frobenius Lie algebras [13].
The construction of the dual cocycle in Section 2 is, however, rather inexplicit and in Section 3 we find a formula for such a cocycle for the semidirect products G = Q V such that V is abelian and the action of Q on the dual abelian group V has a free orbit of full measure (Assumption 2.15). It is well-known that producing a dual 2-cocycle/twist is essentially equivalent to finding a G-equivariant deformation of an appropriate algebra of functions on G. Our assumptions on G = Q V imply that we can identify L 2 (G) with L 2 (V ×V ) in a G-equivariant way. The Kohh-Nirenberg quantization of V ×V provides then a deformation of L 2 (G) and gives rise to a dual unitary 2-cocycle Ω (Theorem 3.12). The cohomology class of this cocycle is exactly the one we defined in Section 2 (Theorem 3.18).
In fact, there are two versions of the Kohn-Nirenberg quantization, so we get two cohomologous dual cocycles. In the case of the ax + b group we show that one of these cocycles coincides with Stachura's cocycle (Proposition 3.28).
Finally, in Section 4 we consider the bicrossed product defined by a matched pair of two copies of Q in Q V . We show that this quantum group is self-dual and isomorphic to (W * (G), ΩΔ(·)Ω * ) (Theorem 4.1 and Corollary 4.2). This is achieved by showing that the multiplicative unitary of the twisted quantum group (W * (G), ΩΔ(·)Ω * ) is given by a pentagonal transformation on Q ×V (Theorem 3.26) and by applying the Baaj-Skandalis procedure of reconstructing a matched pair of groups from such a transformation [2].

Preliminaries
Let G be a locally compact group. We fix a left invariant Haar measure dg on G and denote by L p (G), p ∈ [1, ∞], the associated function spaces.
The modular function Δ = Δ G is defined by the relation Then Δ(g) −1 dg is a right invariant Haar measure on G.
In a similar way, if q ∈ Aut(G), then the modulus |q| = |q| G of q is defined by the identity We let λ and ρ be the left and right regular (unitary) representations of G on L 2 (G): For a function f on G we define a function f by We also let J = J G and Ĵ =Ĵ G be the modular conjugations of L ∞ (G) and W * (G) := λ(G) : Jf :=f andĴf := Δ −1/2f , and we use the shorthand notation The multiplicative unitary Ŵ =Ŵ G of the dual quantum group is defined bŷ The coproduct Δ : W * (G) → W * (G)⊗W * (G) on the group von Neumann algebra W * (G) is defined by Δ (λ g ) = λ g ⊗ λ g . We then havê Let now V be a locally compact Abelian group and V be its Pontryagin dual. Elements of V will be denoted by the Latin letters v, v j , v . . . while elements of V will be denoted by the Greek letters ξ, ξ j , ξ . . . . We will use additive notation both on V and on V .
The duality paring V × V → T will be denoted by e i ξ,v . This is just a notation, we do not claim that there is an exponential function here. We also let We fix a Haar measure dv on V and we normalize the Haar measure dξ of V so that the Fourier transform F V defined by For functions in several variables only one of which is in V , we use the same symbol F V to denote the partial Fourier transform in that variable.

Projective representations and Galois objects
Hopf-Galois objects is a well-studied topic in Hopf algebra theory. An adaption of this notion to locally compact quantum groups has been developed by De Commer [11]. Let us recall the main definitions. We will do this for genuine groups, as this is mainly the case we are interested in, but it will be important for us that the theory is developed at least for locally compact groups and their duals.
Let G be a locally compact group and β be an action of G on a von Neumann algebra N . Such an action is called integrable if the operator-valued weight is semifinite. If β is in addition ergodic, then we get a normal semifinite faithful weightφ on N such that P (a) =φ(a)1. Note that ϕ(β g (a)) = Δ(g) −1φ (a) for all a ∈ N + . (2.1) We can then define an isometric map where Λ : Nφ → L 2 (N , φ) denotes the GNS-map. The pair (N , β) consisting of a von Neumann algebra N and an ergodic integrable action β of G on N is called a G-Galois object if the Galois map G is unitary.
The following characterization of Galois objects is often easier to use.
consisting of a von Neumann algebra N and an ergodic integrable action β of a locally compact group G on N is a G-Galois object if and only if N G is a factor, which is then necessarily of type I.
Proof. This is a simple consequence of results in [11,Section 2]. Indeed, the integrability assumption implies that N G has a canonical representation η on L 2 (N , φ), and then by [11, Theorem 2.1] the pair (N , β) is a G-Galois object if and only if η is faithful. If N G is a factor, then η is faithful, so we get one implication in the proposition. Next, the ergodicity of the action β implies that the action on N by the automorphisms Ad η(λ g ) is ergodic as well, that is, N ∩ η(W * (G)) = C1. It follows that η(N G) = B(L 2 (N , φ)). Hence, if (N , β) is a Galois object, then N G is a factor canonically isomorphic to B(L 2 (N , φ)).
We will be interested in G-Galois objects that are themselves factors of type I, in which case we say, following [12], that (N , β) is a I-factorial G-Galois object. Identifying N with B(H) for a Hilbert space H, we then get a projective unitary representation π : G → P U(H) such that β g = Ad π(g). Note that the equivalence class of π is uniquely determined by (N , β).
Remark 2.2. Ergodicity of β = Ad π is equivalent to irreducibility of π. Assuming ergodicity, integrability of the action Ad π is equivalent to square-integrability of the irreducible projective representation π, meaning that there are nonzero vectors ξ, ζ for which the function g → (π(g)ξ, ζ) is square-integrable. This observation goes back to [9, Example 2.8, Chapter III], but let us give some details.
Note for future use that property (2.1) translates into (Ad π(g))(K) = Δ(g)K. (2.2) Since the action Ad π is ergodic, this determines K uniquely up to a scalar factor. In particular, as was already observed in [14], if G is unimodular then K is scalar, and otherwise K is unbounded.
Remark 2.3. By [11], given a G-Galois object, we get a locally compact quantum group G obtained by reflecting G across the Galois object. If G is abelian, then G = G. But if G is a nonabelian genuine locally compact group and our Galois object has the form (B(H), Ad π) for a projective representation π of G, then G is a genuine quantum group. Indeed, assume G is a group. By the general theory we know that B(H) is a G -Galois object with respect to an action β of G commuting with the action of G, see [11]. There exist scalars χ g (g ) ∈ T such that β g (π(g)) = χ g (g )π(g) for all g ∈ G, g ∈ G , where π(g) is any lift of π(g) to U (H). Then χ g is a character of G . Furthermore, by ergodicity of the action β , if χ g 1 = χ g 2 for some g 1 , g 2 ∈ G, then π(g 1 ) and π(g 2 ) coincide up to a scalar factor, which by surjectivity of the Galois map for (B(H), Ad π) is possible only when g 1 = g 2 . Therefore the map g → χ g is an injective homomorphism from G into the group of characters of G . Hence G is abelian, which contradicts our assumption.
By a recent duality result of De Commer [12], for any locally compact quantum group G, there is a bijection between the isomorphism classes of I-factorial G-Galois object and I-factorial Ĝ -Galois objects. This bijection is constructed as follows. Suppose we are given a I-factorial G-Galois object (N , β). Then N ∩ (N G), equipped with the dual action, becomes a I-factorial Galois object for Ĝ . (More precisely, we rather get a Galois object for the opposite comultiplication on L ∞ (Ĝ) and then an additional application of modular conjugations is needed to really get a Galois object for Ĝ , but this is unnecessary in our setting of genuine groups and their duals.) There is a simple class of Ĝ -Galois objects constructed as follows. Assume from now on that G is second countable. Let ω be a T -valued Borel 2-cocycle on G. Consider the ω-twisted left regular representation of G on λ ω : G → B(L 2 (G)) defined by satisfying λ ω g λ ω h = ω(g, h)λ ω gh , and let W * (G; ω) := λ ω (G) ⊂ B(L 2 (G)). Then W * (G; ω) equipped with the coaction λ ω g → λ ω g ⊗ λ g of G (or in other words, the action of Ĝ ) is a Ĝ -Galois object, see [11,Section 5] for this statement in the setting of locally compact quantum groups.
In fact, this covers all possible I-factorial Ĝ -Galois objects and by duality we get a description of the I-factorial G-Galois objects: Theorem 2.4. For any second countable locally compact group G, there is a bijection between the isomorphism classes of I-factorial G-Galois objects and the cohomology classes [ω] ∈ H 2 (G; T ) such that the twisted group von Neumann algebra W * (G; ω) is a type I factor.
Here H 2 (G; T ) denotes the Moore cohomology of G, which is based on Borel cochains [23].
Explicitly, the bijection in the theorem is defined as follows. To a I-factorial G-Galois object (B(H), Ad π) we associate the cohomology class [ω π ] ∈ H 2 (G; T ) defined by the projective representation π of G. Recall that this means that we lift π to a Borel map π : G → U (H) and define a Borel T -valued 2-cocycle ω π on G by the identitỹ π(g)π(h) = ω π (g, h)π(gh).
For finite groups G, this theorem is essentially due to Wassermann [34] (see also [25]), as well as to Davydov [10] in the purely algebraic setting.
We divide the proof of Theorem 2.4 into a couple of lemmas. Let (B(H), Ad π) be a I-factorial G-Galois object and ω be the cocycle defined by a lift π of π. Lemma 2.5. The Ĝ -Galois object associated with the I-factorial G-Galois object (B(H), Ad π) is isomorphic to (W * (G; ω), α), where the coaction α of G is defined by Proof. We have an isomorphism Explicitly, the crossed product B(H) G is the von Neumann subalgebra of B(L 2 (G; H)) generated by the operators λ g considered as operators on L 2 (G; H) and the operators T for T ∈ B(H) defined by (T ξ)(g) = π(g) * T π(g)ξ(g).
Then the required isomorphism is given by Ad U , where U : This gives the result.
In particular, it follows that W * (G; ω) is a type I factor. As JW * (G; ω)J = W * (G; ω), the von Neumann algebra W * (G; ω) is a type I factor as well. By De Commer's duality result [12] we conclude that the map associating [ω π ] to the isomorphism class of (B(H), Ad π) is injective and its image is contained in the set of cohomology classes [ω] such that W * (G; ω) is a type I factor.
Consider now an arbitrary cohomology class [ω] ∈ H 2 (G; T ) such that W * (G; ω) is a type I factor. To finish the proof it suffices to establish the following.

Dual cocycles
An important class of G-Galois objects arises from dual 2-cocycles. By a dual unitary 2-cocycle on G we mean a unitary element Ω ∈ W * (G)⊗W * (G) such that Similarly to the Ĝ -Galois objects W * (G; ω) considered above, such cocycles lead to G-Galois objects W * (Ĝ; Ω) (which for notational consistency with W * (G; ω) we should have rather denoted by W * (Ĝ; Ω * )). Following [26,Section 4], they can be described as follows.
Identify as usual the Fourier algebra A(G) with the predual of W * (G). Given a dual unitary 2-cocycle Ω ∈ W * (G)⊗W * (G), the von Neumann algebra N := W * (Ĝ; Ω) ⊂ B(L 2 (G)) is generated by the operators Define an action β of G on N by where we remind that ρ g =Ĵλ gĴ is the right regular representation. The map π Ω is the representation of the algebra A(G) equipped with the new product (2.5) The representation π Ω has the equivariance property (2.6) By [33, Section 1.3], the canonical weight φ on N has the following description. Its GNS-space can be identified with L 2 (G), with the GNS-map Λ : Nφ → L 2 (G) uniquely determined byΛ where we remind that f (g) = f (g −1 ). In particular, for f as above we havẽ Two dual unitary 2-cocycles Ω, Ω are called cohomologous if there exists a unitary u ∈ W * (G) such that The cohomology classes form a set H 2 (Ĝ; T ). In general this set does not have any extra structure.
Proposition 2.7. Two dual unitary 2-cocycles Ω, Ω on a locally compact group G are cohomologous if and only if they define isomorphic G-Galois objects.
Denote by G and G the corresponding Galois maps. Then by definition we have On the other hand, by [11,Proposition 5 Using again that Δ (u) =Ŵ * (1 ⊗ u)Ŵ , we conclude that Ω = (u ⊗ u)ΩΔ(u) * .
Combined with Theorem 2.4 this proposition allows one to describe a part of H 2 (Ĝ; T ) in terms of cohomology of G. Namely, denote by H 2 I (Ĝ; T ) the subset of H 2 (Ĝ; T ) formed by the classes [Ω] such that W * (Ĝ; Ω) is a type I factor. Given such an Ω, we can identify W * (Ĝ; Ω) with B(H) for a Hilbert space H. Then the action β of G on W * (Ĝ; Ω) is given by a projective representation π of G on H, and we denote by c Ω the corresponding 2-cocycle ω π on G. Similarly, denote by H 2 I (G; T ) the subset of H 2 (G; T ) formed by the classes [ω] such that W * (G; ω) is a type I factor. A natural question is whether this embedding is onto. We do not know the answer, but as a step towards a solution of this problem let us explain how dual cocycles arise from Galois maps under extra assumptions.
It will be useful to go beyond Galois objects. Assume we are given a square-integrable irreducible projective representation π of G on H. Assume also that we are given a unitary map which we will call a quantization map. Here HS(H) denotes the Hilbert space of Hilbert-Schmidt operators on H. As in Section 2.1, consider the Duflo-Moore operator K on H and the weight φ = Tr(K 1/2 · K 1/2 ). As the GNS-space for φ we could take HS(H), with the GNS-map Λ uniquely determined by Λ (T K −1/2 ) := T for T ∈ HS(H) such that T K −1/2 is a bounded operator. But using the unitary Op we can transport everything to L 2 (G). Thus we take L 2 (G) as the GNS-space, with the GNS-map uniquely determined bỹ Consider the corresponding Galois map G, so G : , where we remind that J = JĴ.
Proposition 2.9. With the above setup and notation, the operator Ω is coisometric. It lies in the algebra W * (G)⊗W * (G) and satisfies the cocycle identity (2.4).
Proof. Since the Galois maps are always isometric, it is clear that Ω is coisometric.
Turning to the cocycle identity, by [11,Proposition 3.5] the Galois map G (denoted by G in op. cit.) satisfies the following hybrid pentagon relation: (More precisely, the result in [11] is formulated only for the Galois objects, but an inspection of the proof shows that it remains valid for arbitrary integrable ergodic actions.) and using Δ (x) =Ŵ * (1 ⊗ x)Ŵ and the pentagon relation Ŵ 12Ŵ13Ŵ23 =Ŵ 23Ŵ12 we obtain the required cocycle identity. Finally, by the definition of Ω, the Galois map G is unitary if and only if Ω is unitary. Assuming that Ω and G are unitary, by [11, Proposition 3.6(1)] the elements (ω ⊗ ι)(G) for ω ∈ B(L 2 (G)) * span a σ-weakly dense subspace of πφ(B(H)). Recalling the definition of W * (Ĝ; Ω) we conclude that The action of G on B(H) is implemented on the GNS-space by the unitaries λ g , while that on W * (Ĝ; Ω) by the unitaries ρ g . Since J ρ g J = λ g , we see that the Galois objects (B(H), Ad π) and (W * (G; Ω), β) are indeed isomorphic.
Remark 2.10. Although [11, Proposition 3.6(1)] is formulated only for the Galois objects, its proof remains valid for any integrable ergodic action. Therefore we see that starting from a square-integrable irreducible projective representation π of G on H and a unitary quantization map Op : L 2 (G) → HS(H), we can define a dual coisometric cocycle Ω by (2.12) and then the set of elements (ω ⊗ ι)(Ŵ Ω * ) for ω ∈ B(L 2 (G)) * span a σ-weakly dense subspace of J πφ(B(H))J . Therefore Ω contains a complete information about (B(H), Ad π) independently of whether we deal with a Galois object or not.
Remark 2.11. The quantization map Op defines a product on L 2 (G) by In general the product Ω defined by Ω is apparently not the same as on A(G) ∩ L 2 (G). In order to see this, let us proceed a bit informally, without trying to fully justify every step. By definition we have Take functions f 1 , f 2 ∈ L 2 (G) and consider the function f ∈ A(G) defined by f (g) = (λ g f 1 , f 2 ). Applying (·f 1 , f 2 ) to the first leg of G(J ⊗ 1) we then get We thus conclude that Ω is related to the quantization map Op defined by The products defined by the quantization maps Op and Op coincide if and only if the is an automorphism with respect to . In general we see no reason why this should be the case. But this can happen. Observe that since Δ is the only positive measurable function F on G such that λ g F = Δ(g) −1 F , any reasonable extension of Op to a class of functions including Δ s should satisfy Op( For the examples studied in this paper we will indeed have such identities, with c = 1 and α = 1/2. On other hand, for the example studied in [5] (which is not a Galois object) we had c = 1 and α = 1/4.
We thus see that the problem of describing H 2 I (Ĝ; T ) reduces to the following question: it is true that for any I-factorial Galois object (B(H), Ad π) there is a unitary quantization map (2.9)? This can be reformulated as a representation-theoretic problem as follows. Assume we are given a 2-cocycle ω on G such that W * (G; ω) is a type I factor. We identify W * (G; ω) with B(H) and put π ω (g) := λ ω g ∈ B(H). Then g → π ω (g) ⊗π c ω (g) is a well-defined unitary representation of G on H ⊗H, where π c ω (g)ξ := π ω (g)ξ. Is this representation equivalent to the regular representation?
The answer is known to be "yes" for finite groups [34,24]. Indeed, the Galois map gives a unitary equivalence where ε L denotes the trivial representation of G on the Hilbert space L. This implies the required equivalence π ω ⊗ π c ω ∼ λ for finite groups G, but falls short of what we need for general G.
Remark 2.12. In order to stress that it can be dangerous to rely too much on analogies with the finite group case, note that for any square-integrable irreducible projective representation π of G on H, the Galois map always defines an embedding of the representation Ad π ⊗ ε HS(H) on HS(H) ⊗ HS(H) into ρ ⊗ ε HS(H) . It follows that for finite groups existence of a unitary quantization map (2.9) is equivalent to (B(H), Ad π) being a Galois object. This is certainly (but until recently unexpectedly!) not the case for general groups. For example, for the connected component G of the ax + b group over R there are two inequivalent infinite dimensional irreducible unitary representations. They are both square-integrable and both admit unitary quantization maps [4]. But G has no I-factorial Galois objects, since H 2 (G; T ) is trivial and W * (G) is the sum of two type I factors.

Dual cocycles defined by genuine representations
We now turn to Galois objects (B(H), Ad π) defined by genuine representations.
Recall that by Proposition 2.7 the cohomology class of Ω is determined by the isomorphism class of the corresponding Galois object. Therefore the above theorem shows that if W * (G) is a type I factor, then we get a canonical class [Ω] ∈ H 2 (Ĝ; T ). In terms of Corollary 2.8, this class corresponds to the unit of H 2 (G; T ).
Note also that the condition that W * (G) is a type I factor implies that G is neither compact nor discrete, so the situation described in the theorem is a purely analytical phenomenon.
Proof of Theorem 2.13. The first statement is an immediate consequence of Theorem 2.4 applied to genuine representations and, correspondingly, to the trivial 2-cocycle ω = 1 on G. Furthermore, that theorem implies that π is unique up to equivalence as a projective representation. Therefore to prove part (i) we only have to show the slightly stronger statement that π is also unique up to equivalence as a genuine representation. Thus, we identify W * (G) with B(H), take π(g) = λ g , and we have to show that for any character η : G → T the representations ηπ and π are equivalent. The representations ηλ and λ are unitarily equivalent, e.g., by Fell's absorption principle. It follows that there exists an automorphism θ of W * (G) such that θ(λ g ) = η(g)λ g for all g. As θ is an automorphism of B(H) = W * (G), it is unitarily implemented, which means exactly that ηπ and π are unitarily equivalent.
In order to prove part (ii), by our results in Section 2.2 it suffices to show that the representation π ⊗ π c is equivalent to the regular representation.
Since we can identify W * (G) with B(H) in such a way that π(g) = λ g , the representation λ is a multiple of π, so we can write λ ∼ π ⊗ ε L , where ε L is the trivial representation on a separable Hilbert space L. Since G is nontrivial, the Hilbert space H must be infinite dimensional. But then the multiplicity of the square-integrable representation π in λ must be infinite as well [14], so the Hilbert space L is infinite dimensional.
By passing to the conjugate representations we get This implies that the irreducible representations π c and π are equivalent. Next, using Fell's absorption principle we get From this we see that the representation π ⊗ π is a multiple of π, and in order to conclude that π ⊗ π ∼ λ it suffices to show that the multiplicity of π in π ⊗ π is infinite. In other words, we have to check that the commutant of (π More generally, let us show that if G 1 is a closed nonopen subgroup of a second countable locally compact group G 2 such that W * (G 1 ) is a type I factor, then the relative is finite dimensional. Denote by Δ i the modular function of G i , by μ i the Haar measure on G i and by φ i the standard Haar weight on W * (G i ). The modular group of φ 2 is given by σ t (λ g ) = Δ 2 (g) it λ g . It preserves W * (G 1 ), and since W * (G 1 ) is a type I factor, there exists a normal semifinite faithful weightφ 1 on W * (G 1 ) with the same modular group. Consider the unique normal semifinite operator-valued weight P : is a type I factor and W * (G 1 ) ∩ W * (G 2 ) is finite dimensional, it follows, e.g., from [32,Corollary 12.12] applied to M = W * (G 1 ) that such an operator-valued weight must be bounded, hence it is a scalar multiple of a conditional expectation. In particular, the Denote by E the conditional expectation obtained by rescaling P , so that c φ 1 E =φ 2 . Using the identity φ 2 (xy) = c φ 1 (E(x)y) for appropriate elements x ∈ W * (G 2 ) and y ∈ W * (G 1 ), it is not difficult to compute E on a dense set of elements. Namely, if Applying this to functions f ≥ 0 supported in arbitrarily small neighborhoods of the identity and normalized by f 1 = 1, which form an approximate unit in L 1 (G 2 ), and using the assumed continuity of E, we conclude that But this formula certainly defines an unbounded map, so we get a contradiction. Indeed, take any nonzero function f ∈ C c (G 1 ). Since G 1 has zero measure in G 2 , we can extend f to a function f ∈ C c (G 2 ) with the same supremum-norm but supported in a set of arbitrarily small Haar measure, so that the L 1 -norm of f can be made arbitrarily small. Since the operator norm is dominated by the L 1 -norm, we thus see that the preimage of G 1f (g)λ g dμ 1 (g) under E has elements of arbitrarily small norm.
As we have already observed, the class [Ω] ∈ H 2 (Ĝ; T ) corresponds to the unit of H 2 (G; T ), so one might wonder whether Ω is also a coboundary, that is, cohomologous to 1. But this is surely not the case, since In fact, the following stronger nontriviality property holds.
Proposition 2.14. If G is a nontrivial second countable locally compact group with group von Neumann algebra a factor of type I, and Ω is the dual unitary 2-cocycle given by Theorem 2.13, then the twisted locally compact quantum group (W * (G), ΩΔ(·)Ω * ) is neither commutative nor cocommutative.
Proof. The algebra W * (G), being a nontrivial type I factor, is clearly noncommutative. This implies that the group G is noncommutative. By Remark 2.3 it follows that the quantum group G Ω obtained by reflecting G across the Galois object (W * (Ĝ; Ω), β) is a genuine quantum group, that is, the coproduct Δ Ω on W * (G Ω ) is not cocommutative. [11].

Examples: subgroups of the affine group
We now introduce the main class of examples studied in this paper. Let V be a nontrivial second countable locally compact abelian group, Q be a second countable locally compact group of continuous automorphisms of V and call G the semidirect product Q V ⊂ Aff(V ) := Aut(V ) V . It has the group law The unit element is (id, 0) and the inverse is (q, v) −1 = (q −1 , −q −1 v). Whenever convenient we identify q ∈ Q with (q, 0) ∈ Q V and v ∈ V with (id, v) ∈ Q V . We will also usually write 1 instead of id for the unit of Q.
We denote by q the dual action of Q on V defined by the identity We will study the groups Q V satisfying the following property.
is a measure class isomorphism. In this case, we say that (V, Q) satisfies the dual orbit condition.
We note in passing that the groups studied in [17] do not satisfy this condition.
Remark 2.16. If (V, Q) satisfies the dual orbit condition, then V cannot be compact, since the neutral element of V cannot belong to the Q-orbit of ξ 0 and therefore V cannot be discrete. Note also that the stabilizer of ξ 0 in Q must be trivial, so that the map φ is injective.
In order to give some concrete examples, let K be a nondiscrete locally compact skewfield. If K is commutative then K is a local field. This means that K is isomorphic either to R, C, a finite degree extension of the field of p-adic numbers Q p or to a field F q ((X)) of Laurent series with coefficients in a finite field F q . If K is a skew-field, then K is isomorphic to a finite dimensional division algebra over a local field k. As an abelian group K is self-dual, with a pairing K × K → T given by (a, b) → χ(Tr K/k (ab)) for a nontrivial character χ of k. Denote by χ K the character χ • Tr K/k . Here also, both (V, Q) and (V , Q) satisfy the dual orbit condition.
Example 2.19. For n ≥ 1 and m ≥ 2, let Then (V, Q) satisfies the dual orbit condition, since for M 1 , . . . , M m , B 1 , . . . , B m−1 ∈ Mat n (K) and A ∈ GL n (K) we have On the other hand, the dual pair (V , Q) does not satisfy the dual orbit condition, as there is no Q-orbit of full measure in V . Let us now say a few words about the case when Q is a real or complex Lie group, V is a finite dimensional vector space and the action of Q on V is given by a representation ρ. In this case we can identify V with V * , and the action of Q on V * is given by the contragredient representation ρ c . If (Q, V ) satisfies the dual orbit condition, then Q has the same dimension as V and the map φ : Q → V * is open, so (V * , ρ c (Q)) is a prehomogeneous vector space.
Remark 2.21. In the complex case, assuming dim Q = dim V , the dual orbit condition is satisfied for ξ 0 ∈ V * as long as ξ 0 has trivial stabilizer. Indeed, consider the set Ω of vectors ξ ∈ V * such that the map q → V * , X → (dρ c )(X)ξ, is a linear isomorphism. This set is nonempty, as ξ 0 ∈ Ω, and Zariski open in V * . It follows that it is a dense, connected, open subset of V * in the usual topology. Since the Q-orbit of every element of Ω is open, it follows that Ω consists entirely of one orbit, so the dual orbit condition is satisfied. As a byproduct we see that Q must be connected.
Note that these arguments do not apply in the real case, as a Zariski open subset of R n can have finitely many connected components.
By a repeated use of the following simple lemma we can construct more and more complicated examples. Proof. By Remark 2.21 it suffices to find a vector η 0 ∈ g * with trivial stabilizer. By assumption there exists ξ 0 ∈ V * with trivial stabilizer in Q. Identifying g with q × V we let η 0 (X, w) := ξ 0 (w). The adjoint action is given by Assume now that (q, v) ∈ G lies in the stabilizer of η 0 , that is, Taking X = 0, we see that q stabilizes ξ 0 , hence q = e. Since the map q → V * , X → (dρ c )(X)ξ 0 , is an isomorphism, we then conclude that v = 0.
This lemma and its proof show, both in real and complex cases, that if (Q, V ) satisfies the dual orbit condition, then the Lie algebra of G := Q V is Frobenius, meaning that G has an open coadjoint orbit, or equivalently, there exists η 0 ∈ g * such that the bilinear form η 0 ([X, Y ]) on g is nondegenerate. This has already been observed in [28]. Moreover, the converse is almost true: by [28,Theorem 4.1], if dim Q = dim V and the Lie algebra of Q V is Frobenius, then there exists ξ 0 ∈ V * such that the map q → V * , X → (dρ c )(X)ξ 0 , is a linear isomorphism. This is a bit less than what we need since the stabilizer of ξ 0 can still be a nontrivial discrete subgroup of Q and, in the real case, the open set ρ c (Q)ξ 0 ⊂ V * can be nondense.
In addition to Lemma 2.22, another way of producing new examples is to start with a pair (Q, V ) satisfying the dual orbit condition and multiply the representation of Q on V by a quasi-character of Q. This can destroy the dual orbit condition, but not necessarily.
Example 2.23. Consider Q = C * , V = C, with Q acting on V by multiplication. Obviously, the dual orbit condition is satisfied. By Lemma 2.22 the adjoint action of the ax + b group G := Q V satisfies the dual orbit condition as well. This is basically Example 2.19 for n = 1 and m = 2. If we multiply the adjoint representation of G by the quasi-character (q, v) → q k , k ∈ Z, we get the representation The action of G on C 2 defined by ρ k has a dense open dual orbit if and only if k = −1. The dual orbit condition is satisfied if and only if k = 0 or k = −2, while for other k = −1 the stabilizers of the points on the dense dual orbit are finite and nontrivial. We can use the same formulas in the real case. Then the dual orbit condition is satisfied if and only if k is even.
We remark in passing that for k = 1 the group G ρ 1 C 2 is an extension of C * by the Heisenberg group. It is used in the construction of an extended Jordanian twist in [20].
Returning to the general case, we have the following result. By Theorem 2.13 it follows that we get a canonical class [Ω] ∈ H 2 (Ĝ; T ) defined by the Galois object (W * (G), Ad λ). In order to get an explicit representative of this class we need to fix a (unique up to equivalence) representation π as in that theorem and choose a unitary quantization map (2.9). This is the task we are going to undertake in the next section.

Kohn-Nirenberg quantization of V ×V
Quantization of the abelian self-dual group V ×V has been considered long ago [31,35].
Here we consider the Kohn-Nirenberg quantization. It has some benefit compared with the Weyl type quantization for which one has to assume the map V → V , v → 2v, to be a homeomorphism.
The Kohn-Nirenberg quantization can initially be defined as the continuous injective linear map from tempered Bruhat distributions on V ×V to continuous linear operators from the Bruhat-Schwartz space on V to tempered Bruhat distributions on V (see [7,Section 9] for a precise definition). It is defined by the formula Remark 3.1. This quantization map should rather be called anti-Kohn-Nirenberg. The Kohn-Nirenberg one is given by the formula The two quantization maps are unitarily equivalent and for our purposes it is easier to work with anti-Kohn-Nirenberg.
The distributional kernel of the operator Op KN (F ) is therefore given by Since an operator on L 2 (V ) with kernel K is Hilbert-Schmidt if and only if K ∈ L 2 (V × V ), we immediately deduce: Hence, the Hilbert space L 2 (V ×V ) can be endowed with an associative product Very important here are the symmetries of the Kohn-Nirenberg product 0 . We have an action of Aff(V ) on V ×V : With | · | V the modulus function of Aut(V ) and | · |V the one of Aut(V ), we observe that We will usually use only the modulus |q| V in various formulas and write it simply as |q|. The action (3.3) gives rise to a unitary representation π V ×V of Aff(V ) on L 2 (V ×V ) defined by We can also define a unitary representation π V of Aff(V ) on L 2 (V ) by In particular, the operators of the representation π V ×V of Aff(V ) act on L 2 (V ×V ) by algebra automorphisms for 0 .
Proof. Take ϕ ∈ L 2 (V ). Then On the other hand, Using that and that the Haar measure dw dξ of V ×V is invariant under the transformations (w , ξ) → (qw , q ξ), we can write the last integral as and this is equal to (3.6) by translation invariance of the Haar measure dw on V .

Dual cocycles on subgroups of Aff(V )
Let Q V ⊂ Aff(V ) be a subgroup satisfying the dual orbit Assumption 2.15. Let d Q (q) be a Haar measure on Q and Δ Q the modular function. Routine computations show that a left invariant Haar measure and modular function on G are respectively given by The measure class isomorphism: We therefore may assume that the Haar measure on Q is normalized so that the G-equivariant linear operator Ũ φ : where φ(q) = q ξ 0 , is unitary. Equivalently, our normalization is such that the Haar measure dξ onV is the push-forward under φ of the measure |q| −1 d Q (q) on Q, that is, Note in passing that this uniquely determines the Haar measure on G: if we multiply dv by a scalar, then (3.9) and unitarity of F V force us to divide d Q (q) by the same scalar. Therefore we get a unitary quantization map and if we denote by π the canonical representation of G ⊂ Aff(V ) on L 2 (V ), that is, π = π V | G , then by Lemma 3.3 we have Op(λ g f ) = π(g) Op(f )π(g) * .
Lemma 3.4. The representation π of G on L 2 (V ) is irreducible, square-integrable and the associated Duflo-Moore formal degree operator is Since Δ is a V -invariant function on G, it is naturally viewed as a function on Q. Therefore by M (Δ −1 • φ −1 ) we mean the operator of multiplication by the function Proof. It is more convenient to work on L 2 (Q) with the equivalent representation π given byπ The restriction of the representation π to V is simply the regular representation. It follows that any operator in B(L 2 (V )) commuting with π(G) must belong to π(V ) . Passing to the equivalent representation π, this means that any operator in B(L 2 (Q)) commuting with π(G) must be an operator of multiplication by a function in L ∞ (Q). In addition this function must be invariant under the left translations on Q, hence it is constant. Thus π is irreducible.
Turning to square-integrability, for ϕ 1 , ϕ 2 ∈ C c (Q), we have with the second equality following from (3.9). If we set then, for every fixed q ∈ Q, the function f q has compact essential support and is bounded.
The Plancherel formula for V gives then This shows that π is square-integrable and the Duflo-Moore operator is the operator of multiplication by the function q → |q|/Δ Q (q). This gives the result.
The next natural question is whether (B(L 2 (V )), Ad π) is a G-Galois object, or equivalently, by Theorem 2.13 and Proposition 2.24, whether π is quasi-equivalent to the regular representation. If it is, then we can construct a dual unitary 2-cocycle on G by the procedure described in Section 2.2. Instead of exactly following that procedure, however, we will construct this cocycle directly from the product on L 2 (G) defined by According to Remark 2.11 this approach should not necessarily work, but if it does, it has some technical advantages.
Let us start by observing that by definition the algebra (L 2 (G), ) is unitarily isomorphic to the algebra HS(L 2 (V )) of Hilbert-Schmidt operators. Hence (3.12) We will need explicit formulas for the product on dense subspaces of (L 2 (G), ). First, we introduce an auxiliary space: Definition 3.5. Let E(G) be the Banach space completion of C c (G) with respect to the norm Lemma 3.6. For any f 1 , f 2 ∈ E (G) and a.a. (q, v) ∈ G, we have

13)
and Here, following our conventions, F V : L 2 (G) = L 2 (Q V ) → L 2 (Q ×V ) is the partial Fourier transform in V -variables. Note that for this map to be unitary we have to equip Q ×V with the measure |q| −1 d Q (q) dξ (which in general is not the Haar measure of the semidirect product Q V for the dual action).
Proof. For f ∈ L 2 (G), we let K f ∈ L 2 (V × V ) be the kernel of the Hilbert-Schmidt operator Op(f ). From (3.1) and (3.10) we get If f 1 , f 2 ∈ E(G), then, since Op(f 1 f 2 ) = Op(f 1 )Op(f 2 ), the product formula for operator kernels gives where the last step is justified by Fubini's theorem: note that for any belongs to L 1 (V ×V ×V ) and its L 1 -norm is not greater than , w), (3.14) with absolutely converging integrals. It is easy to see that f ∈ L 2 (G) and In particular, we can compute K f , the kernel of the operator Op(f ): which by Fubini (and a simplification of the phases) becomes Hence K f 1 f 2 = K f , and (3.13) follows by injectivity of the map To get the second formula in the formulation of the lemma, we apply the partial Fourier transform to (3.14) and using Fubini's theorem one more time obtain This concludes the proof.
Proof. Since C c (G) ⊂ E(G), the result follows from formula (3.14) which clearly entails that when f 1 , f 2 ∈ C c (G), then also f 1 f 2 ∈ C c (G).
Next we consider a space of functions with good behavior in the partial Fourier space: Definition 3.8. For a measure space (X, μ), we let L(X, μ) be the subspace of L ∞ (X, μ) consisting of functions that are (essentially) zero outside a set of finite measure. We then let FL(G) be the subspace of L 2 (G) consisting of functions of the form a subalgebra of (L 2 (G), ). Moreover, for any f 1 , f 2 ∈ FL(G) and  a.a. (q, ξ) ∈ Q ×V , we have Proof. Let K j ⊂ Q and K j ⊂V (j = 1, 2) be Borel sets of finite measure such that f j is (essentially) zero outside K j ×K j . Then the function defined by the right hand side of (3.15) is zero for (q, ξ) outside K 1 × (K 1 +K 2 ). Therefore if (3.15) holds, then f 1 f 2 ∈ FL(G).
Turning to the proof of (3.15), we already know from Lemma 3.6 that this identity holds for (3.16) for almost all (q, ξ). By (3.12) we have h 1,n h 2,n → f 1 f 2 in L 2 (G) as n → ∞, hence the left hand side of (3.16) (considered as a function in (q, ξ)) converges to F V (f 1 f 2 ) in the L 2 -norm. On the other hand, the right hand side converges to the right hand side of (3.15) by the dominated convergence theorem. Therefore to finish the proof it suffices to show that for every f ∈ FL(G) there exists a sequence of functions h n ∈ E(G) ∩FL(G) First for all, if (K n ) n is an increasing sequence of compact subsets of Q ×V with union Q ×V , then F * Let K be any compact such that its interior contains the support of F V f . By Lusin's theorem, we can find a uniformly bounded sequence of continuous functions g n supported in K such that g n → F V f a.e. Then F * V (g n ) ∈ FL(G) and In a similar fashion, by approximating functions in C c (Q ×V ) by elements of the algebraic tensor product C c (Q) ⊗ C c (V ), we may assume that f = F * V (g ⊗ h) = g ⊗ F * V h for some g ∈ C c (Q) and h ∈ C c (V ). Finally, by approximating h by the convolution of two functions, we may assume that f = g ⊗ F * V (h 1 * h 2 ) for g ∈ C c (Q) and h i ∈ C c (V ). But such a function is already in E(G).
It follows from (3.13) that a candidate for the dual cocycle on G defining the product by formula (2.5) is given by (q, v). (3.17) For the moment this is just a formal expression, but it at least makes sense as a sesquilinear formΩ on C c (G × G): Our first goal is to prove that Ω makes sense as a unitary operator on L 2 (G × G). This will be proven by showing that Ω factorizes as a product of three unitaries.
Next, by the definition of Ω it is clear that Ω commutes with the operators ρ g ⊗ 1 and 1 ⊗ ρ g . Hence Ω ∈ W * (G)⊗W * (G).
Proof. Recall that A(G) consists of functions of the form ϕ 1 * φ 2 , with ϕ i ∈ L 2 (G), which correspond to the linear functionals (· ϕ 2 , φ 1 ) on W * (G). Under the identification Using the initial definition of Ω as a bilinear form on C c (G × G), we get Hence the equality in the formulation of the lemma holds for all f 1 , f 2 of the form ϕ 1 * φ 2 , with ϕ i ∈ C c (G). Therefore in order to prove the lemma it suffices to show that every function f ∈ A(G) ∩ L 2 (G) can be approximated by functions of the form ϕ 1 * φ 2 , with ϕ i ∈ C c (G), simultaneously in the norms on A(G) and L 2 (G).
Consider first a function of the form f = f 1 * f 2 , with f 1 ∈ L 2 (G) and f 2 ∈ C c (G). If ϕ n → f 1 in L 2 (G), ϕ n ∈ C c (G), then ϕ n * f 2 → f 1 * f 2 both in A(G) and L 2 (G).
Consider now an arbitrary f ∈ A(G) ∩ L 2 (G). By the previous case in order to finish the proof it suffices to show that f can be approximated simultaneously in A(G) and L 2 (G) by functions of the form f * φ, with ϕ ∈ C c (G). Take a standard approximate unit (ϕ n ) n in L 1 (G) consisting of functions ϕ n ∈ C c (G) such that ϕ n ≥ 0, G ϕ n dg = 1, with the supports of ϕ n eventually contained in arbitrarily small neighborhoods of the unit. Then f * φ n → f in L 2 (G). At the same time, if we write f as f 1 * f 2 for some f i ∈ L 2 (G) and use that We thus see that A(G) ∩ L 2 (G) is a subalgebra of (L 2 (G), ). Since A(G) ∩ L 2 (G) is dense in A(G), the associativity of the product on this subalgebra implies that Ω satisfies the cocycle identity (2.4).
To summarize, we have proved the following result. Remark 3.13. If we started from the opposite Kohn-Nirenberg quantization (see Remark 3.1) we would have obtained the following dual 2-cocycle: which differs from Ω by the inversion of legs and by the sign of the phase: where R is the unitary antipode of W * (G) given on the generators by R (λ g ) = λ g −1 .
By [11, Proposition 6.3, iii)], the dual cocycles Ω and Ω are cohomologous, with the unitary operator implementing the cohomological relation equal to J J, which we will explicitly compute in Section 3.5.

Identification of the Galois objects
To complete the picture it remains to check that the Galois object defined by the dual cocycle Ω is exactly the pair (B(L 2 (V )), Ad π).
For f ∈ L 2 (G), consider the operator L (f ) on L 2 (G) defined by By (3.12) we have Furthermore, under the identification of (L 2 (G), ) with HS(L 2 (V )) via Op, the map L is simply the left regular representation of HS(L 2 (V )) on itself. This representation is a multiple of the canonical representation of HS(L 2 (V )) on L 2 (V ). It follows that there is a unique isomorphism Therefore in order to find an isomorphism W * (Ĝ; Ω) ∼ = B(L 2 (V )) it suffices to find an isomorphism W * (Ĝ; Ω) ∼ = L (L 2 (G)) . From formula (2.7) for the GNS-map on W * (Ĝ; Ω), for every f ∈ A(G) we have the following equality: π Ω (f )ϕ = S L (f ) Sϕ for all ϕ ∈ A(G) ∩ L 2 (G) such that the right hand side is well-defined, where S is the unbounded operator defined by Sϕ =φ. In other words, using the unitary operator J defined in (1.1), for all ϕ ∈ A(G) ∩ L 2 (G) such that the right hand side is well-defined.
We thus see that we need to understand a connection between the operators L (f ) and M (Δ s ). For this we will first get another useful formula for L . First, we denote by U φ the variant of the unitary operator Ũ φ , defined in (3.8), without the permutations of variables: (3.21) Then we denote by γ the unitary representation of V on L 2 (G) given by where τ ξ : L 2 (V ) → L 2 (V ) is the left regular representation of V given by (τ ξ ϕ)(ξ ) = ϕ(ξ − ξ).
Next, for fixed ξ ∈V and f ∈ FL(G), we denote by (F V f )(•, ξ) the V -invariant function on G given by (q, v) → (F V f )(q, ξ) and by M (F V f )(•, ξ) the bounded operator on L 2 (G) of multiplication by the function (F V f )(•, ξ). Lemma 3.9 then leads to the following result.
Lemma 3.14. For any f ∈ FL(G), we have the absolutely convergent (in the operator norm) integral formula Finally, we introduce a family (T z ) z∈C of operators on functions on G by where we remind that Δ(q ) = Δ Q (q )/|q |. We need a dense subspace of FL(G) preserved by these operators: We denote by L 0 (Q ×V ) the union of the spaces L K,L (Q ×V ) and by FL 0 (G) (respectively, by FL K,L (G)) the subspace of FL(G) consisting of functions f ∈ FL(G) such that F V f belongs to L 0 (Q ×V ) (respectively, to L K,L (Q ×V )).
Proof. By definition (3.24), the operator T z conjugated by the partial Fourier transform is the operator of multiplication by the function This immediately shows that FL K,L (G) is stable under T z as the modular function Δ is bounded, as well as bounded away from zero, on any compact subset of Q. Hence FL 0 (G) is also stable under T z .
Since F V is unitary, to prove density of FL 0 (G) in L 2 (G) it suffices to show that For this it suffices to show that for every compact set K ⊂ Q the union of the sets (3.25) over the compact sets L ⊂ Q is a subset of K ×V of full measure. But this is clear, since this union is and by assumption φ(Q) is a subset of V of full measure.
Since the map f →f is bounded on , the functions f for f ∈ FL 0 (G) form a dense subspace of L 2 (G) as well. Hence the functions ϕ * f for ϕ ∈ C c (G) and f ∈ FL 0 (G) are dense in A(G). An easy calculation shows that for any g = (q, v) ∈ G and compacts K, L ⊂ Q, we have λ g (FL K,L (G)) ⊂ FL qK,L (G). Hence, if f ∈ FL K,L (G) and ϕ ∈ C c (G) has support contained in U ×V for a compact set U ⊂ Q, then ϕ * f ∈ FL UK,L (G). Therefore Taking a standard approximate unit in L 1 (G) for ϕ, we see also that functions of the form ϕ * f , for ϕ ∈ C c (G) and f ∈ FL 0 (G), are dense in FL 0 (G), hence in L 2 (G). Moreover, since the operators T z are bounded on the spaces FL K,L (G), we may also conclude that the functions T z (ϕ * f ) are dense in L 2 (G) for all z. In particular, Proposition 3.17. The operator M (Δ) is affiliated with the von Neumann algebra L (L 2 (G)) . Moreover, for all f ∈ FL 0 (G) and z ∈ C, we have Proof. Since Δ depends only on the coordinate Q, the operators M (Δ it ), t ∈ R, commute with the partial Fourier transform F V . From formula (3.15) we see then that As L (FL 0 (G)) is σ-weakly dense in L (L 2 (G)) , this implies that M (Δ) is affiliated with L (L 2 (G)) . The same formula also shows that since Next, formula (3.23) and definition (3.24) of T z give, for f ∈ FL 0 (G), the absolutely convergent integral On the other hand, using again (3.23), we have on Δ (z) L 2 (G): As F V commutes with operators of multiplication by V -invariant functions, from this we easily deduce: Hence we have which by closedness of M (Δ −z ) finally gives This proposition and identity (3.20) imply that for any f ∈ A(G) ∩ FL 0 (G) we have , hence on L 2 (G), as both sides of the identity are bounded operators.
is dense in L 2 (G) by Lemma 3.16, it follows that W * (Ĝ; Ω) = J L (L 2 (G)) J . Recalling also that the action of G on W * (Ĝ; Ω) is given by the automorphisms Ad ρ g , we see that this action transforms under the isomorphism Ad J of W * (Ĝ; Ω) onto L (L 2 (G)) into the action given by the automorphisms Ad λ g . Using the isomorphism (3.19) the latter action transforms, in turn, into the action Ad π on B(L 2 (V )). We therefore get the required isomorphism of the Galois objects: Theorem 3.18. For any second countable locally compact group G = Q V satisfying the dual orbit Assumption 2.15, the G-Galois object (W * (Ĝ; Ω), β) defined by the dual cocycle (3.17) is isomorphic to (B(L 2 (V )), Ad π); explicitly, the isomorphism maps π Ω (f ), As a byproduct we see that (B(L 2 (V )), Ad π) is indeed a Galois object. We remind that by Theorem 2.13 this is therefore the unique up to isomorphism I-factorial Galois object defined by a genuine representation of G.
We are therefore exactly in the situation discussed in Remark 2.11, with identities (2.13) satisfied for c = 1 and α = 1/2. This "explains" why we were able to construct the dual cocycle Ω using the quantization map Op instead of the modified quantization map Op .

Deformation of the trivial cocycle
We continue to consider a second countable locally compact group G = Q V satisfying the dual orbit Assumption 2.15 and a dual unitary 2-cocycle Ω on G defined by (3.17).
By replacing the distinguished point ξ 0 ∈V by ξ q 0 := q ξ 0 we in fact get a family of such dual cocycles Ω q indexed by the elements q ∈ Q. We already know that all these cocycles are cohomologous, since they correspond to the unique Galois object defined by a genuine representation. This is also easy to see as follows.
Proposition 3.20. We have Ω q = (λ q ⊗ λ q )ΩΔ(λ q ) * for all q ∈ Q. In particular, the map q → Ω q is continuous in the so-topology.
Proof. Considering as before the partial Fourier transform F V as a map From this and Lemma 3.10 we get On the other hand, consider the map φ q : Q →V defined by ξ q 0 = q ξ 0 , so that φ q (q ) := q ξ q 0 = φ(q q). Then which shows that Ξ q is exactly the map from the expression for Ω q in Lemma 3.10 (with ξ 0 replaced by ξ q 0 ). This proves the proposition.
Although the dual cocycles Ω q are all cohomologous, under quite general assumptions they can be used to construct a continuous deformation of the trivial cocycle. Namely, we have the following result. Proof. Using the notation from the proof of the previous proposition it suffices to show that Ξ z −1 n → id a.e., or equivalently, φ −1 (ξ 0 + z n ξ) → e for a.e. ξ ∈V . But this is clear, since by assumption the image of φ contains a neighborhood of ξ 0 and φ is a homeomorphism of Q onto its image.
Example 3.22. Consider the ax + b group G over the reals, so that Q = R * , V = R, and Q acts on V by multiplication. In other words, G is the group of matrices We identify R with R via the pairing e ixy . Then s t = s −1 t for s ∈ Q and t ∈V . Take −1 as ξ 0 . We then get a continuous family of cohomologous dual unitary 2-cocycles Ω θ , θ ∈ R * , on G such that In this case the pointwise convergence θ → 0 in End(V ) means that θ −1 → 0 in R. So by the above proposition we have Ω θ → 1 as θ → 0, which is obviously the case.

Multiplicative unitaries
Our next goal is to find an explicit formula for the multiplicative unitary of the twisted quantum group (W * (G), ΩΔ(·)Ω * ) for the dual cocycle Ω defined by (3.17). The main issue is to determine the modular conjugation J for the canonical weightφ on W * (Ĝ; Ω).
It is more convenient to work with the isomorphic Galois object (L (L 2 (G)) , Ad λ). The algebra (L 2 (G), ) becomes a * -algebra if we transport the * -structure on HS(L 2 (V )) to L 2 (G). Since the * -structure on the algebra of Hilbert-Schmidt operators is isometric, the corresponding * -structure on L 2 (G) must have the form f → UJf for a unitary operator U on L 2 (G). In other words, the operator U is defined by the identity Op(f ) * = Op(UJf).
Lemma 3.23. The operator U is given by where U φ is the operator defined by (3.21).
Proof. Consider the unitary Op :

Then by definition Op(f ) is the integral operator with kernel
. In other words, Using that we arrive at Finally, using that U φ =Ũ φ Σ and we get the desired formula. Proof. The proof is similar to that of [5,Proposition 2.8]. Let us start with the canonical weightφ L for the Galois object (L (L 2 (G)) , Ad λ). Note that since M (Δ) is affiliated with L (L 2 (G)) and the function Δ is, up to a scalar factor, the only positive measurable function F on G such that λ g F = Δ(g) −1 F , the isomorphism L (L 2 (G)) ∼ = B(L 2 (V )), where K is the Duflo-Moore operator of formal degree (explicitly given by Lemma 3.4). We have densely defined operators on HS(L 2 (V )) of multiplication on the right by c z K −z (z ∈ C). Correspondingly, we have densely defined operators on L 2 (G), which we suggestively denote by f → f Δ z . Thus, by definition, for f in a dense subspace of L 2 (G). Explicitly, by Proposition 3.17 we have Recalling the description of the GNS-representation for (B(L 2 (V )), Ad π) in Section 2.2, and formula (2.10) in particular, we see that as the GNS-space for φ L we can take L 2 (G), with the GNS-map Λ L : Nφ L → L 2 (G) uniquely determined bỹ Λ L (L (f )) = c 1/2 f Δ −1/2 for f ∈ L 2 (G) such that the right hand side is well-defined. The corresponding modular conjugation J L is simply given by the involution on L 2 (G) ∼ = HS(L 2 (V )), so J L = UJ. Now, using the isomorphism Ad J between W * (Ĝ; Ω) and L (L 2 (G)) , we can consider the space L 2 (G) as the GNS-space for φ using the map Nφ x → JΛ L (J xJ ). (3.28) In this picture the modular conjugation for φ is J UJJ . Therefore we only have to check that the above GNS-map is exactly the map Λ used to define J . Recall that Λ is given byΛ for all such f . By comparing the norms of both sides we can already conclude that c = 1. Then, since any two GNS-representations associated with φ are unitary conjugate and the vectors Λ (π Ω (f )) =f , with f ∈ A(G) ∩ FL 0 (G), form a dense subspace of L 2 (G), it follows that the maps Λ and (3.28) are equal.
As was shown in [11], the unitary operator J J must belong to W * (G). The following makes this explicit.
Lemma 3.25. For any ϕ ∈ C c (G) and g ∈ G we have: Proof. We have, with absolutely convergent integrals: As J ρ g J = λ g , applying this to J ϕ instead of ϕ we get the announced formula.
By [11,Proposition 5.4], we deduce that the multiplicative unitary Ŵ Ω for the deformed quantum group (W * (G), ΩΔ(·)Ω * ) is given by the formulâ (3.29) Conjugating by the partial Fourier transform, we get a more explicit formula:  (3.17). Then for the multiplicative unitary Ŵ Ω of the deformed quantum group (W * (G), ΩΔ(·)Ω * ) and any f ∈ L 2 (G × G) we have Proof. We know by Lemma 3.10 that for f ∈ L 2 (Q ×V × G, From this we obtain, for f ∈ L 2 (G × G): It follows that On the other hand, with the help of Lemma 3.25 we can perform calculations similar to those of Lemma 3.10 to get, for f ∈ L 2 (Q ×V , |q| −1 d q (q) dξ), that Moreover, one easily finds that Hence we get Using that the expression above becomes which is what we need.
Recall that in Section 3.4 we considered a continuous family of cohomologous dual unitary 2-cocycles Ω q , q ∈ Q, defined by replacing ξ 0 by ξ q 0 = q ξ 0 .
Corollary 3.27. We have: Q →V is open and z n → 0 in End(V ) pointwise for a sequence of elements Proof. Part (i) follows already from formula (3.29). Indeed, the map q → Ω q is continuous by Proposition 3.20. On the other hand, the unitary U q in formula (3.29) for the dual cocycle Ω q is given by Lemma 3.23, with ξ 0 replaced by ξ q 0 . To be more precise, that lemma is formulated under the assumption that the Haar measure on Q is normalized by (3.9). If we replace ξ 0 by ξ q 0 = q ξ 0 , and, correspondingly, the map φ by φ q (q ) = φ(q q), but want to keep the same measure on Q, then the map is unitary only up to a scalar factor. But this means that for Lemma 3.23 to remain true we just have to state it as the equality As the map q → U φ q is obviously continuous in the so-topology, we conclude that the map q → U q is continuous as well, hence so is the map q →Ŵ Ω q .
Since by (3.30) we have Y = (F V ⊗ F V )Ŵ (F * V ⊗ F * V ), this proves the result.

Stachura's dual cocycle
In this section we consider the simplest example of our setup, the ax + b group G over the reals. In this case Stachura [30] already defined a dual cocycle on G. We refer the reader to his paper for a motivation of the construction and just present an explicit form of the cocycle.
Consider the following operators affiliated with W * (G): Then the dual unitary cocycle on G found by Stachura, see [30,Lemma 5.6], is defined by Proposition 3.28. The dual cocycle Ω S coincides with the dual cocycle Ω defined by (3.18) for ξ 0 = −1. In particular, Ω S is cohomologous to the dual cocycle Ω defined by (3.17).
Consider now, for fixed s ∈ R, the function ϕ(y) := s + ln |1 + y|. It has two simple zeros located at y ± = ±e −s − 1 and it is continuously differentiable (away from −1) with ϕ (y ± ) = ±e s . Therefore Setting now q = e s and v = −t we get Setting ξ 0 = −1 and remembering that we have here q ξ = q −1 ξ and d(q, v) = (2π) −1 q −2 dq dv, we finally get which is exactly the dual cocycle Ω defined by (3.18) in Remark 3.13. As we already observed there, Ω is cohomologous to Ω.
As was suggested by Stachura [30], the quantum group (W * (G), Ω SΔ (·)Ω * S ) is isomorphic to the quantum ax + b group of Baaj and Skandalis [29] (see also [33,Section 5.3]), but his arguments fall a bit short of proving that this is indeed the case. The above proposition together with Theorem 4.1 below complete his work.

Bicrossed product construction
Recall that a pair (G 1 , G 2 ) of closed subgroups of a locally compact second countable group G is called a matched pair if G 1 ∩ G 2 = {e} and G 1 G 2 is a subset of G of full measure [3]. Given such a pair, we have almost everywhere defined measurable left actions α of G 1 and β of G 2 on the measure spaces G 2 and G 1 , resp., such that gs −1 = α g (s) −1 β s (g) for g ∈ G 1 , s ∈ G 2 .
We can then define a bicrossed product Ĝ 1 G 2 . This is a locally compact quantum group with the function algebra The coproduct on L ∞ (Ĝ 1 G 2 ) is a bit more difficult to describe, but we will not need to know the exact definition and refer the reader for that to [3] or [33]. 3 Then by [33, Proposition 2.9 and Theorem 2.13] the dual quantum group is G 1 Ĝ 2 =Ĝ 2 G 1 , the bicrossed product defined by the matched pair (G 2 , G 1 ) of subgroups of G.
In a similar way there exist a (unique up to isomorphism) second countable locally compact group G 2 , a right action of G 2 on X and an equivariant measurable map f 2 : X → G 2 such that for almost all pairs (x, y) ∈ X × X we have x * y = yf 2 (x).
In our case we get According to [2,3] we then get a locally compact group G and embeddings h i : G i → G of the groups G i as closed subgroups such that the map G 1 × G 2 → G , (s, g) → h 1 (s)h 2 (g) is injective and the complement of its image is a set of measure zero. By [2, Lemma 3.5(b)], for almost all (x, y) ∈ X × X we have h 2 (f 2 (x))h 1 (f 1 (y)) = h 1 (f 1 (b))h 2 (f 2 (a)), where (a, b) = v(x, y).
Corollary 4.2. The locally compact quantum group (W * (G), ΩΔ(·)Ω * ) is self-dual. If G is nontrivial, then this quantum group is noncompact and nondiscrete, and if G is nonunimodular (that is, Δ Q = | · | V ), then the quantum group is also nonunimodular, with nontrivial scaling group and scaling constant 1.