Symmetry, Integrability and Geometry: Methods and Applications Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs ⋆

We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in $SL(2,\mathbb{C}$) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a $\mathbb{Z}_2$-grading, we obtain product formulae for little $q$-Jacobi functions.


Introduction
In the classical setting of locally compact groups, a Gelfand pair consists of a locally compact group G, together with a compact subgroup K such that the convolution algebra of bi-Kinvariant L 1 -functions on G is commutative. See [5] or [8] for a comprehensive introduction. Gelfand pairs give rise to spherical functions and a spherical Fourier transform which decomposes bi-K-invariant functions on G as an integral of spherical functions, see [5,Theorem 6.4.5] or [8,Théorème IV.2].
For many examples, this decomposition is made precise [5]. The examples include the group of motions of the plane together with its diagonal subgroup and the pair (SO 0 (1, n), SO(n)), where SO 0 (1, n) is the connected component of the identity of SO (1, n). In particular the spherical functions are determined and one can derive product formulae for these type of functions.
Since the introduction of quantum groups, Gelfand pairs were studied in a quantum context, see for example [9,26,39,40] and also the references given there. These papers consider pairs of quantum groups that are both compact. For such pairs it suffices to stay with a purely (Hopf-)algebraic approach. Under the assumption that every irreducible unitary corepresentation admits only one matrix element that is invariant under both the left and right action of the subgroup, these quantum groups are called (quantum) Gelfand pairs. Classically, this is equivalent to the commutativity assumption on the convolution algebra of bi-K-invariant elements. If the matrix coefficients form a commutative algebra one speaks of a strict (quantum) Gelfand pair. In the group setting every Gelfand pair is automatically strict and as such strictness is a purely non-commutative phenomenon.
For quantum groups, many deformations of classical Gelfand pairs do indeed form a quantum Gelfand pair that moreover is strict. As a compact example, (SU q (n), U q (n − 1)) forms a strict Gelfand pair [40]. In a separate paper [38] Vainerman introduces the quantum group of motions of the plane, together with the circle as a subgroup as an example of a Gelfand pair of which the larger quantum group is non-compact. As a result a product formula for the Hahn-Exton q-Bessel functions, also known as 1 ϕ 1 q-Bessel functions, is obtained [38,Corollary,p. 324], see also [16,Corollary 6.4]. However, a comprehensive general framework of quantum Gelfand pairs in the non-compact operator algebraic setting was unavailable at that time.
At the turn of the millennium, locally compact (l.c.) quantum groups have been put in an operator algebraic setting by Kustermans and Vaes in their papers [21,22], see also [19,31,42]. The definitions give a C * -algebraic and a von Neumann algebraic interpretation of locally compact quantum groups. Many aspects of abstract harmonic analysis have found a suitable interpretation in this von Neumann algebraic framework. In particular, Desmedt proved in his thesis [4] that there is an analogue of the Plancherel theorem, which gives a decomposition of the left regular corepresentation of a l.c. quantum group.
From this perspective, it is a natural question if the study of Gelfand pairs can be continued in the l.c. operator algebraic setting. In this paper we give this interpretation. Motivated by Desmedt's proof of the quantum Plancherel theorem, we define the necessary structures to obtain a classical Plancherel-Godement theorem [5,Theorem 6.4.5] or [8,Théorème IV.2]. For this the operator algebraic interpretation of Gelfand pairs is essential.
We keep the setting a bit more general than one would expect. For a classical Gelfand pair of groups, one can prove that the larger group is unimodular from the commutativity assumption on bi-K-invariant elements. Here we will study pairs of quantum groups for which the smaller quantum group is compact and we assume that the larger group is unimodular. We will not impose the classically stronger commutativity assumption. The reason for this is that we would like to study SU q (1, 1) ext together with its diagonal subgroup. However, the natural analogue of the commutativity assumption would exclude this example.
We mention that it is known that the notion of a quantum subgroup is in a sense too restrictive. Using Koornwinder's twisted primitive elements, it is possible to define double coset spaces associated with SU q (2) and get so called (σ, τ )-spherical elements, see [15] for this particular example. See also [14] for a similar study of SU q (1, 1) on an algebraic level. The subgroup setting then corresponds to the limiting case σ, τ → ∞. In the present paper we do not incorporate such a general setting.
Motivated by the Hopf-algebraic framework, we introduce the non-compact analogues of bi-K-invariant functions and its dual [9,40] and equip these with weights. We will do this in a von Neumann algebraic manner and for the dual structure also in a C * -algebraic manner. We prove that the C * -algebraic weight lifts to the von Neumann algebraic weight. Moreover, we establish a spherical analogue of a theorem by Kustermans [18] which establishes a correspondence between representations of the (universal) C * -algebraic dual quantum group and corepresentations of the quantum group itself. Eventually, this structure culminates in a quantum Plancherel-Godement theorem, as an application of [4,Theorem 3.4.5]. This illustrates the advantage of an operator algebraic interpretation above the Hopf algebraic approach. In particular, we get a spherical L 2 -Fourier transform, or spherical Plancherel transformation, and we show in principle that this is a restriction of the non-spherical Plancherel transformation.
As an example, we treat the first example of a quantum Gelfand pair involving a q-deformation of SU (n, 1). Namely, we treat the quantum analogue of the normalizer of SU (1, 1) in SL(2, C), which we denote by SU q (1, 1) ext , see [13] and [10]. We identify the circle as its diagonal subgroup and study the spherical properties of this pair. We see that the classical commutativity assumption on the convolution algebra is too restrictive to capture SU q (1, 1) ext with its diagonal subgroup. Nevertheless, the pair exhibits properties reminiscent of classical Gelfand pairs. In particular, we see how one can derive product formulae using gradings on this quantum group and its dual.
We mention that the q-deformation of SU (1, 1) was first established on the operator algebraic level in [13]. The construction heavily relies on q-analysis. More recently, de Commer [3] was able to obtain SU q (1, 1) ext using Galois co-objects. So far, the higher dimensional q-deformations of SU (n, 1) remain undefined on the von Neumann algebraic level.

Structure of the paper
In Section 3 we study the homogeneous space of left and right invariant elements. We also give their dual spaces. Main goal here is the introduction of the von Neumann algebras N andN as well as the C * -algebrasN c andN u . These are homogeneous counterparts of the von Neumann algebra of a quantum group and its dual as well as the underlying reduced and universal dual C * -algebra.
In Section 4, we study the natural weights on these homogeneous spaces. Since the weight on the larger quantum group is generally not a state, the analysis is much more intricate compared to the compact Hopf-algebraic approach. We prove that the C * -algebraic weights defined here lift to the von Neumann algebraic (dual) weight. This result is a major ingredient for the quantum Plancherel-Godement theorem, since it allows us to apply Desmedt's auxiliary theorem [4,Theorem 3.4.5]. Next, we introduce the necessary terminology of corepresentations that admit a vector that is invariant under the action of a subgroup. This is worked out in Section 5.
In Section 6 we elaborate on a spherical version of Kustermans' result [18]: there is a 1-1 correspondence between representations of the universal dual of a quantum group and corepresentations of the quantum group itself. This will form the essential bridge between [4, Theorem 3.4.5] and the quantum Plancherel-Godement Theorem 7.1. Eventually, Section 7 combines the results of Sections 3-6 to prove a quantum version of the Plancherel-Godement theorem.
In Section 8 we work out the example of SU q (1, 1) ext together with its diagonal subgroup. We determine all the objects defined in Sections 3-7. As an application of the theory we find product formulae for little q-Jacobi-functions that appear as matrix coefficients of irreducible corepresentations.

Preliminaries and notation
We briefly recall the definition and essential results from the theory of locally compact quantum groups. The results can be found in [21,22] and [18]. For an introduction we refer to [31] and [19]. For the theory of weights on von Neumann algebras we refer to [30].
We use the following notational conventions. If π and ρ are linear maps, we write πρ for the composition π • ρ. ι denotes the identity homomorphism. The symbol ⊗ will be used for either the tensor product of two elements, of linear maps, the von Neumann algebraic tensor product or the tensor product of representations. We use the leg-numbering notation for operators. For example, if W ∈ B(H) ⊗ B(H), we write W 23 for 1 ⊗ W and W 13 = (Σ ⊗ 1)W 23 (Σ ⊗ 1), where Σ : H ⊗ H → H ⊗ H is the flip. For a linear map A, we denote Dom(A) for its domain.
Let B be a Banach * -algebra. With a representation of B, we mean a * -homomorphism from B to the bounded operators on a Hilbert space, which is referred to as the representation space. If B is a C * -algebra, we write Rep(B) for the equivalence classes of representations of B and IR(B) for the equivalence classes of irreducible representations of B. With equivalence, we mean unitary equivalence. With slight abuse of notation we sometimes write π ∈ Rep(B) (or π ∈ IR(B)) to mean that π is an (irreducible) representation of B instead of looking at its class.
If M is a von Neumann algebra and ω ∈ M * , we denoteω for the normal functional defined byω(x) = ω(x * ). For ω ∈ M * and x, y ∈ M , we denote xω, ωy, xωy for the functionals defined by (xω Let φ be a weight on M . Let n φ = {x ∈ M | φ(x * x) < ∞}, m φ = n * φ n φ . Let σ φ be the modular automorphism group of φ. We denote T φ for the Tomita algebra defined by For x, y ∈ T φ , we write xφy for the normal functional determined by (xφy)(z) = φ(yzx), z ∈ M .

Quantum groups
We use the Kustermans-Vaes definition of a locally compact quantum group [21,22], see also [19,31,42]. 3. Two normal, semi-finite, faithful weights ϕ, ψ on M so that ϕ is the left Haar weight and ψ the right Haar weight.
Note that we suppress the Haar weights in the notation. A locally compact quantum group (M, ∆) is called compact if ϕ and ψ are states. (M, ∆) is called unimodular if ϕ = ψ. Compact quantum groups are unimodular. The triple (H, π, Λ) denotes the GNS-construction with respect to the left Haar weight ϕ. We may assume that M acts on the GNS-space H. We use J and ∇ for the modular conjugation and modular operator of ϕ and σ for the modular automorphism group of ϕ. Recall that there is a constant ν ∈ R + called the scaling constant such that ψσ t = ν −t ψ. By applying [34], we see that there is a positive, self-adjoint operator δ, called the modular element, such that ψ = ϕ δ , i.e. formally ψ(·) = ϕ(δ 1 2 · δ 1 2 ). For compact quantum groups, the scaling constant and the modular element are trivial.

Multiplicative unitary
There exists a unique unitary operator W ∈ B(H ⊗ H) defined by

The unbounded antipode
To (M, ∆) one can associate an unbounded map called the antipode S : Dom(S) ⊆ M → M . It can be defined as the σ-strong- * closure of the map (ι ⊗ ω)(W ) → (ι ⊗ ω)(W * ), where ω ∈ B(H) * . One can prove that there exists a unique * -anti-automorphism R : M → M and a unique strongly continuous one-parameter group of * -automorphisms τ : R → Aut(M ) such that R is called the unitary antipode and τ is called the scaling group. Moreover, where Σ M,M : M ⊗ M → M ⊗ M is the flip. Using the relative invariance property of the left Haar weight with respect to the scaling group, we define P to be the positive operator on H such that P it Λ(x) = ν t 2 Λ(τ t (x)), t ∈ R, x ∈ n ϕ . We use the notation In that case ω * (x) = ω(S(x)), x ∈ Dom(S). For ω ∈ M ♯ * , we set ω * = max{ ω , ω * }. M ♯ * becomes a Banach- * -algebra with this norm.
Corepresentations A (unitary) corepresentation is a unitary operator U ∈ M ⊗ B(H U ) that satisfies the relation (∆ ⊗ ι)(U ) = U 13 U 23 . In this paper all corepresentations are assumed to be unitary. It follows from the pentagon equation that the multiplicative unitary W is a corepresentation on the GNSspace H. Two corepresentations U 1 , U 2 are equivalent if there is a unitary T : We denote IC(M ) for the set of equivalence classes of all unitary corepresentations. For any corepresentation U ∈ M ⊗ B(H U ) and ω ∈ B(H U ) * , (ι ⊗ ω)(U ) ∈ Dom(S) and

Reduced quantum groups
To every locally compact quantum group (M, ∆) one can associate a reduced C * -algebraic quantum group. We define M c to be the norm closure of {(ι⊗ω)(W ) | ω ∈M * }, which is a C * -algebra. We restrict ∆, ϕ and ψ to M c and denote the respective restrictions by ∆ c , ϕ c , ψ c . In fact, ∆ c should be considered as a map into the multiplier algebra of the minimal tensor product of M c with itself. The GNS-construction representation (H, Λ, π) then restricts to a GNSrepresentation of the C * -algebraic weight ϕ c , which is denoted by (H, Λ c , π c ). It is proven in [21] that (M c , ∆ c ) forms a reduced C * -algebraic quantum group. Similarly, one defines the reduced dual C * -algebraic quantum group (M c ,∆ c ). The associated objects are denoted with a hat.

Universal quantum groups
Universal quantum groups were introduced by Kustermans [18]. For ω ∈ M ♯ * , we define ω u = sup{ π(ω) | π a representation of M ♯ * }. Recall that with a representation, we mean a * -homomorphism to the bounded operators on a Hilbert space. Note that this defines a norm since the representation λ is injective. LetM u be the completion of M ♯ * with respect to · u . We let λ u : M ♯ * →M u denote the canonical embedding.M u carries the following universal property: if π is a representation of M ♯ * , then there is a unique representation ρ ofM u such that π = ρλ u . In particular, from the representation λ we get a surjective mapθ :M u →M c . We define a universal weight onM u by settingφ u =φ cθ , andψ u =ψ cθ . The GNS-representation ofφ u is given by ( : ω → (ω ⊗ι)(U ) determines a representation ofM u , which we denote by π U . In fact, it is shown in [18] that this establishes a 1-1 correspondence between corepresentations of M and non-degenerate representations ofM u .
For completeness, we mention thatM u can be equipped with a comultiplication∆ u , which is a map fromM u to the multiplier algebra of the minimal tensor product ofM u with itself, such that (M u ,∆ u ) is a universal C * -algebraic quantum group in the sense of [18]. Similarly, one defines M u , ∆ u , ϑ, ϕ u , ψ u , . . ..

Spherical Fourier transforms
Let (G, K) be a pair consisting of a locally compact group G and a compact subgroup K. If L 1 (K\G/K), the bi-K-invariant L 1 -functions on G equipped with the convolution product is a commutative algebra, then (G, K) is a called a Gelfand pair. Examples of Gelfand pairs can be found in [5,Chapter 7] and [8].
A notion of Gelfand pairs for compact quantum groups was introduced by Koornwinder [17]. We briefly recall the definition. Consider two unital Hopf-algebras H, H 1 . Denote ∆ for the comultiplication of H, denote ϕ 1 for the Haar functional on H 1 . Suppose that there exists a surjective morphism π : H → H 1 , so that H 1 is identified as a quantum subgroup of H. Now consider the left and right coactions ∆ l = (π ⊗ ι)∆, ∆ r = (ι ⊗ π)∆. Define Now, the following definition characterizes a quantum Gelfand pair. In fact, there are more equivalent definitions. We state the one which is closest to the theory we develop in the present section. A pair of compact groups (G, K) is a Gelfand pair if and only if the Hopf-algebra of matrix coefficients of unitary finite dimensional representations form a quantum Gelfand pair (which is automatically strict). Many deformations of classical Gelfand pairs form strict Gelfand pairs in the Hopf-algebraic setting. Examples can be found in for instance [9,15,24,39] and [40].
The aim of this section is to give a general framework of Gelfand pairs in the locally compact quantum group setting as introduced by Kustermans and Vaes [21,22]. This puts the earlier studies as mentioned in the introduction in a non-compact, von Neumann algebraic setting.
One of the main motivations for the operator algebraic approach is that we can define spherical Fourier transforms. In particular, we show that the structure presented here allows us to prove a decomposition theorem analogous to the classical Plancherel-Godement theorem [8,Théorème IV.2] or [5,Section 6]. The proof is an application of Desmedt's auxiliary result [4,Theorem 3.4.5].
As explained in the introduction, we do not assume a natural quantum analogue of the classical commutativity assumption on the convolution algebra of bi-invariant functions. Instead, we assume unimodularity of the larger quantum group, which is classically a weaker assumption. This allows us to cover the example of SU q (1, 1) ext , see Section 8.
Notation 3.2. Throughout Sections 3-7, we fix a locally compact quantum group (M, ∆) together with a closed quantum subgroup (M 1 , ∆ 1 ) which we assume to be the compact. Recall [37,Definition 2.9] that this means that we have a surjective * -homomorphism π : M u → (M 1 ) u on the level of universal C * -algebras and the induced dual * -homomorphismπ : (M 1 ) u →M u lifts to a map on the level of von Neumann algebrasπ :M 1 →M , which with slight abuse of notation is denoted byπ again. When we encounterπ in this paper, we always mean the von Neumann algebraic map.
Note π andπ are in principal also used for the GNS-representations of M andM . However, we omit the maps most of the time, since M andM are identified with their GNS-representations. In that case we explicitly need the GNS-representations, we will mention this.
We mention that from a certain point, see Notation 3.17, we will assume that (M, ∆) is a unimodular quantum group.
Proof . For the left hand side, using the pentagon equation and [36, Proposition 3.1], For the right hand side, using again [36, Proposition 3.1], The lemma follows by the fact that the elements {(ι ⊗ ω)(W ) | ω ∈M * } are σ-strong- * dense in M . We recall from [35] that we have normal, faithful operator valued weights, Since (M 1 , ∆ 1 ) is compact, T β and T γ are finite. We extend the domains of T β and T γ to M in the usual way. We denote the extensions again by T β and T γ . The composition of T β and T γ forms a well-defined map on M . Note that . Moreover, we stretch that T β and T γ are conditional expectation values, which properties have been studied in the related papers [28] and [32].
(2) We prove that ∆T γ = (T γ ⊗ ι)∆, the other equation can be proved similarly using β = (R ⊗ ι)γR and (2.1). We find: Now, the equation follows by taking slices of the second leg of W .
Note that ∆ ♮ is unital, but generally not multiplicative.
Remark 3.11. Note that ∆ ♮ is the von Neumann algebraic version of∆ [40], which was used to define hypergroup structures. See also the remarks at the beginning of this section. Here, we will not focus on hypergroups for two reasons. First of all, we stay mostly at the measurable von Neumann algebraic setting, which does not allow one to directly define the generalized shift operators [40]. Moreover, we will not assume that N is Abelian, i.e. what is called a strict Gelfand pair in [40].
Proposition 3.14 defines an orthogonal projectionπ((ϕ 1 ⊗ ι)(W 1 )) for which we simply write Classically, it corresponds to projecting onto the space of functions that are left invariant with respect to the compact subgroup. Note that P γ ∈M and We need a similar result as Proposition 3.14 for T β . For this we need unimodularity of (M, ∆). Classically, if G is a group with compact subgroup K such that (G, K) forms a Gelfand pair, one can prove that G is unimodular, see [8,Proposition I.1]. The natural definition of a quantum Gelfand pair would be to require that ∆ ♮ is cocommutative. However, we like to stretch the definition of a Gelfand pair a bit to handle the example of SU q (1, 1) ext . The following essential result, see Proposition 3.16, is the motivation of assuming the (classically) weaker condition that (M, ∆) is unimodular, see Notation 3.17.
First, we need the following lemma. Note that for a, b ∈ T ϕ , we have aϕb ∈ M * and hence T ϕ ϕT ϕ is a subset of M * . Recall that for ω ∈ I, ξ(ω) ∈ H is defined using the Riesz theorem as the unique vector such that ξ(ω), Λ(x) = ω(x * ), x ∈ n ϕ . By [21,Lemma 8.5], T ϕ ϕT ϕ is included in I.
Proof . For ω ∈ I we define the norm ω I = max{ ω , ξ(ω) }. We have to prove that T ϕ ϕT ϕ is dense in I with respect to this norm. This is exactly what is obtained in the proof of [1,Proposition 3.4]. Indeed, let L and k be as in [1]. As indicated in the introduction of [1], ) is bounded and it extends continuously to the projectionĴP γĴ .
We will write P β for the projectionĴP γĴ . In particular P β ∈M ′ . Under the assumption that (M, ∆) is unimodular, we see that P β projects onto the elements that are right invariant with respect to the closed quantum subgroup (M 1 , ∆ 1 ). Since we will need this interpretation of P β , i.e. Proposition 3.16, we assume unimodularity from now on. We are ready to define the dual structures associated with N . We define left and right invariant analogues of the dual von Neumann algebraic quantum group and the universal dual C * -algebraic quantum group. These duals can be constructed by means of the multiplicative unitary W associated with (M, ∆). We define For ω ∈ N ♯ * , we set ω * = max{ ω , ω * }. Then, N ♯ * becomes a Banach- * -algebra with respect to this norm. Proposition 3.13 shows that ω ∈ N ♯ * if and only ifω ∈ M ♯ * . Note that N ♯ * is dense in N * . Indeed, the restriction map M * → N * : ω → ω| N is continuous and the image of the Using this together with Propositions 3.12 and 3.13, we see that we see thatN is a von Neumann algebra if considered as acting on P γ H, so that P γ is its unit. In particularN = P γM P γ . We defineN c to be the norm closure of the set For ω ∈ N ♯ * , we define Note that this defines a norm since the representation ω → (ω ⊗ ι)(W ) is injective as follows using the bijective correspondence established in Proposition 3.12. LetN u be the completion of N ♯ * with respect to · ♮ u . Recall the map λ : M ♯ * →M : ω → (ω ⊗ ι)(W ). We set Note that the image of λ ♮ is contained inN c and we will use this implicitly. λ u : M ♯ * →M u is the canonical inclusion and similarly Recall thatN u is a C * -algebra with the following universal property: if π is a representation of N ♯ * on a Hilbert space, then there is a unique representation ρ ofN u such that π = ρλ ♮ u . By this universal property, the map N ♯ * →M u : ω → λ u (ω) extends to a representation Similarly, the map λ ♮ : N ♯ * →N c gives rise to a surjective map In particular,θι u =θ ♮ , whereθ :M u →M c was the canonical projection induced by the representation λ : M ♯ * →M c .
Remark 3.18. Note that we do not claim that ι u :N u →M u is injective. In fact, this is generally not true, see the comments made in Remark 6.4 and the paragraph before this remark.

Weights on homogeneous spaces
We introduce weights on the von Neumann algebras and C * -algebras as were introduced in Section 3. We study their GNS-representations and prove Proposition 4.7, which is essential for implementing [4,Theorem 3.4.5].
Recall that the C * -algebraic weightsφ u ,φ c were defined in Section 2. The weights on the von Neumann algebras M ,M and the C * -algebrasM c ,M u restrict to weights on N ,N andN c ,N u by setting respectively. We prove that ϕ ♮ andφ ♮ are normal, semi-finite, faithful weights andφ ♮ c andφ ♮ u are lower semi-continuous, densely-defined, non-zero weights. Here the assumption made in Notation 3.17 becomes essential.
We refer to [21, Section 1.1] for the definition of a GNS-representation for a C * -algebraic weight.
c is a proper (i.e. densely defined, lower semi-continuous, non-zero) weight onN c . Its GNS-representation is given by Proof . By Proposition 4.2, {(ω ⊗ ι)(W ) | ω ∈ I N } ⊆N is a norm dense subset ofN c contained in nφ. The lower semi-continuity and non-triviality follow sinceφ ♮ c is a restriction of the faithful weightφ c . The claim on the GNS-representation follows exactly as in the proof of Proposition 4.3.
Proof . Lemma 4.5 shows that I ∩ M ♯ * is dense in M ♯ * . Since I N consists of the restrictions to N of functionals in I, see Proposition 4.2, and N ♯ * consists of the restrictions to N of functionals in M ♯ * , see Proposition 3.13, it follows that Thus,φ ♮ u is densely defined. Thatφ ♮ u is lower semicontinuous follows from [21, Definition 1.5]. Take any ω ∈ N ♯ * such thatω = 0, which exists since all functionals in N * are given by restrictions of functionals in M * , see Proposition 3.12.
Next, we like to prove thatφ ♮ is essentially the W * -lift ofφ ♮ u , see [21,Definition 1.31]. A priori this question is ill-defined, since these weights are defined on different von Neumann algebras. Indeed,φ ♮ is a weight onN , which by definition acts on P γ H, whereas the W * -lift ofφ ♮ u is a weight on (π u ι u (N u )) ′′ . Since P β ∈M ′ , we see that P γ P β H is an invariant subspace ofN . By Proposition 4.6, the von Neumann algebra (π u ι u (N u )) ′′ equals the restriction ofN to P β P γ H, The point is that thatN andN P β are in fact isomorphic. This follows in fact from Proposition 4.3, but we give a different argument here. We claim, more precisely, that the map N →N P β : x → xP β is an isomorphism. Indeed, for any x in the center ofM ,ĴxĴ = x. Since P β =ĴP γĴ , every projection in the center ofM majorizes P β if and only if it majorizes P γ . Therefore, the central supports of P β and P γ are equal. It follows from [11, Theorem 10.3.3] thatN is isomorphic to P βN P β =N P β , where the isomorphism is given by the map N →N P β : x → xP β .

Spherical corepresentations
We introduce the necessary terminology for corepresentations that admit vectors that are invariant under the action of a quantum subgroup. These corepresentations can be considered as spherical corepresentations.
We denote Note that H M 1 U is a closed subspace of H U . We denote IC(M, M 1 ) for the equivalence classes of irreducible corepresentations of M that admit non-trivial M 1 -invariant vectors. We refer to such corepresentations as spherical corepresentations. If {(ω ⊗ ι)(U )H M 1 U | ω ∈ M * } is dense in H U , then we call U homogeneously cyclic. It should be clear that every irreducible corepresentation of M that admits a non-trivial M 1 -invariant vector is homogeneously cyclic.
Proof . This follows from the following series of equalities for which we use Lemma 3.7 (3) and We mention the following two propositions in order to compare our framework with the setting of classical Gelfand pairs of groups. The proof of the first one is completely analogous to the proof of [8,Proposition II.6] Hence, x is a character of the convolution algebra N ♯ * . It can be considered as a quantum spherical function or quantum spherical element. The equality (5.1) allows one to derive product formulae as is done in for example [38,39,40]. Here we keep to a more general setting and do not assume thatN is Abelian in order to include the example of SU q (1, 1) ext in Section 8. : ω → (ω ⊗ ι)(U ). By the universal property ofN u , we see that this gives rise to a representation ofN u on H M 1 U . Of course, this representation can be trivial. If U is homogeneously cyclic, then the corresponding representation ofN u is non-degenerate. Indeed, suppose that there exist w ∈ H M 1 U , such that for all v ∈ H M 1 U and ω ∈ M ♯ * , (ω ⊗ ι)(U )v, w = 0. Then, using (4) of Lemma 3.7, We see that for all v ∈ H M 1 U and ω ∈ M ♯ * , (ω ⊗ ι)(U )v, w = 0, which proves that w = 0, since U is homogeneously cyclic. Hence, every non-zero homogeneously cyclic corepresentation U of M gives rise to a non-degenerate representation ofN u . Definition 5.7. Let U ∈ M ⊗ B(H U ) be a homogeneously cyclic corepresentation of M on a Hilbert space H U . Then, we get a representation π ♮ U ofN u determined by We emphasize that the representation Hilbert space of π ♮ U is H M 1 U .
Remark 5.8. Let U ∈ M ⊗ B(H U ) be a homogeneously cyclic corepresentation of M . U is irreducible if and only if π ♮ U is irreducible. Indeed, if U is reducible, then clearly π ♮ U is reducible. The only if part follows from a computation similar to the one in Remark 5.6.
Recall that we denote π U for the representation ofM u given by λ u (ω) → (ω ⊗ ι)(U ). Then, π ♮ U equals the restriction of π U ι u to H M 1 U . We will need the following result for Theorem 7.1. We refer to [6] for the theory of direct integration and the definition of a fundamental sequence.
Proposition 5.9. Let X be a measure space, with standard measure µ. Suppose that for every x ∈ X, we have a homogeneously cyclic corepresentation U x of M on a Hilbert space H x such that (π ♮ Ux ) x∈X is a µ-measurable field of representations ofN u . Suppose thatM u is separable. Then, (H x ) x∈X is a µ-measurable field of Hilbert spaces such that (H M 1 x ) x∈X is a µ-measurable field of subspaces and (U x ) x∈X is a µ-measurable field of corepresentations.
We claim that (f i,j x ) x∈X is a fundamental sequence for (H x ) x∈X . Indeed, since U x is homogeneously cyclic, the span of (ω i ⊗ ι)(U x )e j x , i, j ∈ N is dense in H x . Moreover, where the third equality follows by a computation similar to the one in Remark 5.6. Since (π ♮ Ux ) x∈X is a µ-measurable field of representations, we see that (5.2) is a µ-measurable function of x. Hence (f i,j x ) x∈X is a fundamental sequence. Moreover, by a similar computation as (5.2), for any ω ∈ M ♯ * , 2} are equivalent. Then U 1 and U 2 are equivalent.
Proof . Let T : H M 1 U 1 → H M 1 U 2 be the unitary intertwiner between π 1 and π 2 . Let Q 0 be the mapping This map is well-defined and isometric. Indeed, for ω ∈ M ♯ * and v ∈ H M 1 U 1 , where the second equality follows from a similar calculation as in Remark 5.6. Since U 1 and U 2 are homogeneously cyclic, Q 0 is densely defined and has dense range. Let Q : where the second equality follows again from a similar calculation as in Remark 5.6. Since U 1 is homogeneously cyclic, this proves that Q intertwines U 1 with U 2 .
Note that the converse of the previous proposition is clear: if U 1 and U 2 are equivalent corepresentations, then the corresponding representations as considered in Remark 5.6 are equivalent. In this section, we compare the representations of the C * -algebras defined in Sections 2 and 3. Main objective is to prove that the representations ofN c 'lift' to representations ofM c .
Let us give a more elaborate discussion. There are three special types of representations within Rep(N u ).
1. As explained in Remark 5.6, the corepresentations of M give rise to representations ofN u . Recall [18] that the corepresentations of M are in 1-to-1 correspondence with non-degenerate representations ofM u . Hence, the representations ofM u give rise to representations ofN u . This correspondence can be described more directly: if π is a representation ofM u on a Hilbert space H π , then πι u is the corresponding representation ofN u on the closure of ((πι u )(N u ))H π . Note that by Remark 5.11, this assignment descends to a well-defined, injective map on the equivalence classes of representations, 2. If π is a representation ofN c , then πθ ♮ is a representation ofN u . These representations correspond to the representations that are weakly contained in the GNS-representation ofφ ♮ u . Indeed this follows, since by Proposition 4.6, this GNS-representation is given bŷ Hence, every representation πθ ♮ , with π ∈ Rep(N c ) is weakly contained in the GNSrepresentation ofN u . The other way around, any representation ofN u that is weakly contained in the GNS-representation ofφ ♮ u factors through the canonical projectionθ ♮ . 3. If π is a representation ofM c , then πθ ♮ = πθι u is a representation ofN u . Here, we used thatN c ⊆M c .
The main results of this section will be the following. We prove that every representation ofN c comes from a representation ofM c , i.e. the representations ofN u obtained in (2) and (3) are the same ones. Theorem 6.1. For every non-degenerate representation ρ ∈ Rep(N c ), there exists a nondegenerate representation π ∈ Rep(M c ) on a Hilbert space H π such that ρ is equivalent to the restriction of π|N c to the closure of π(N c )H π .
Note that the completion of (N ♯ * ) ∼ ⊗N c H ρ is a closed subspace of the Hilbert space X ⊗N c H ρ that is isomorphic to H ρ via the unitary extension T of The map (6.4) extends unitarily since ρ is non-degenerate. For ω, θ ∈ N ♯ * , v ∈ H ρ , so that (N ♯ * ) ∼ ⊗ H ρ is an invariant subspace for Ind ρ. We denote its closure by Y . Moreover, for ω, θ ∈ N ♯ * and v ∈ H ρ , so that T intertwines (Ind ρ)|N c restricted to the Hilbert space Y with ρ. Finally, we claim that Y equals the closure of (Ind ρ)(N c )X. Indeed, for any ω ∈ N ♯ * , θ ∈ M ♯,β * and v ∈ H ρ , we see that SinceN cNc ⊆N c is dense, it is straightforward to prove that Y equals the closure of (Ind ρ)(N c )X in X. This concludes the proof by choosing π = Ind ρ. Corollary 6.2. For every representation ρ ofN u that factors throughθ ♮ , there is a homogeneously cyclic corepresentation U of M such that ρ is equivalent to π ♮ U . Remark 6.3. An essential ingredient for the proof of the quantum version of the Plancherel-Godement theorem is to see to which corepresentation the GNS-map ofN u corresponds. Recall that the GNS-representation ofφ ♮ u was given by the triple (P γ P β H,Λ u ι u ,π u ι u ), see Proposition 4.6. Sinceπ u ι u =π|N cθ ♮ , we can apply Corollary 6.2. We define the closed subspace where the closure is with respect to the norm in H. It is clear that the representation ofN u that corresponds to the restriction of W to E equalsπ u ι u .
We use the notation IR(M u , M 1 ) to denote the irreducible representations π ofM u such that the representation πι u is non-trivial. Under the 1-1 correspondence between IR(M u ) and IC(M ), see [18], we see from Remark 5.11 and the remarks following Definition 5.7 that IR(M u , M 1 ) corresponds to IC(M, M 1 ). Let IR(M c , M 1 ) denote the irreducible representations ofM c such that the restriction toN c is non-trivial. We find the following diagram of inclusions.
Note that the map IR(M u , M 1 ) ֒→ IR(N u ) indeed maps into the representations ofN u that are irreducible, c.f. Remark 5.8. Hence, also the vertical inclusion on the right hand side of (6.6) preserves irreducibility.
The example in Section 8 shows that the inclusion IR(M c , M 1 ) ֒→ IR(M u , M 1 ) is not surjective. We briefly comment on the fact that also the inclusion IR(M u , M 1 ) ֒→ IR(N u ) is generally not surjective. This is a consequence of the fact that there are Lie groups G with compact subgroup K for which the there are non-unitary representations whose restriction to the bi-Kinvariant functions forms a representation, i.e. a homomorphism that preserves the * -operation, whereas the representation of all L 1 -functions on G does not preserve the * . This happens for example for SL(2, R), see [44, Example 1.1.2 on p. 37 and p. 40]. This shows that the induction argument contained in the proof of Theorem 6.1 does not work in general on the universal level.
Remark 6.4. Assume the map ι u :N u →M u to be injective. Then by general C * -algebra theory, it is isometric. With this additional assumption Theorem 6.1 holds on the universal level. So for every ρ ∈ Rep(N u ), there is a representation π ∈ Rep(M u ) on a Hilbert space H π , such that ρ is equivalent to the restriction of πι u to (πι u (N u ))H π .
The proof is completely analogous to the one of Theorem 6.1, where one takes the universal norm instead of the reduced norm on N ♯ * . The injectivity of ι u plays an essential role at two places. First of all, the injectivity of ι u can be used to prove positivity of the inner product (6.2), since in this case an element inN u is positive if and only if it is positive inM u . Secondly, the universal analogue of (6.3) can be recovered from the injectivity of ι u , since for ω ∈ M ♯ * and θ ∈ M ♯,β * , The rest of the prove of Theorem 6.1 can be copied mutatis mutandis.

A quantum group analogue of the Plancherel-Godement theorem
Here we prove a decomposition theorem that may be considered as a locally compact quantum group version of the Plancherel-Godement theorem as can be found in [8,Théorème IV.2]. The proof is different from the one given in [8] and follows the line of the Plancherel theorem as proved by Desmedt [4]. We show that the C * -algebraN u together with the weight ϕ ♮ u that we introduced and studied so far fit into the framework of [4,Theorem 3.4.5]. Then we use Theorem 6.1 to translate the results in terms of corepresentations of M that admit a M 1 -invariant vector.
Recall that we defined the space E in (6.5). Let L be any Hilbert space and let L 0 ⊆ L be a closed subspace. We denote the conjugate Hilbert space of L by L. Note that the space L ⊗ L 0 can canonically be identified with the Hilbert-Schmidt operators in B(L 0 , L). We denote the latter space by B 2 (L 0 , L). For results on direct integration, we refer to [6,23] and [25]. In particular, we will use [23, Theorem 1.10] implicitly several times. If A and B are unbounded operators such that AB is closable, we denote A · B for the closure of AB.
2. For ω 1 , ω 2 ∈ I N , we have Proof . By Propositions 4.6 and 4.7,φ ♮ u is a proper approximate KMS-weight. Therefore, we can apply [4,Theorem 3.4.5], so that we obtain a measure µ M 1 on IR(N u ), a measurable field of Hilbert spaces (K M 1 σ ) σ∈IR(Nu) , a measurable field of representations (π σ ) σ∈IR(Nu) , a measurable field of self-adjoint, strictly positive operators (D M 1 σ ) σ∈IR(Nu) and an isomorphism Q M 1 0 of P γ P β H onto ⊕ IR(Nu) K M 1 σ ⊗ K M 1 σ dµ M 1 (σ) satisfying the properties of this theorem. Let ρ ∈ IR(N u ) be in the support of µ M 1 . We claim that π ρ is weakly contained inπ u ι u . Suppose that this is not the case, so that there exists x ∈N u such that π ρ (x) = 0 butπ u ι u (x) = 0.

Assume moreover thatM is a type I von Neumann algebra and thatM
(1). It follows from Corollary 6.2, Remark 5.6 and the properties of D M 1 σ described in (1) of [4,Theorem 3.4.5], that for ω ∈ I N , the operator (ω ⊗ ι)(U )(D M 1 U ) −1 is bounded and its closure is Hilbert-Schmidt for µ M 1 -almost every U ∈ IC(M, M 1 ).
(2) and (3). We make two observations. First note that by Proposition 4.6, for ω ∈ I N , Λ u ι u (λ ♮ u (ω)) = ξ(ω). Secondly, we have proved that for every ρ in the support of µ M 1 , there is a U ∈ IC(M, M 1 ), such that π ρ = π ♮ U . Then (2) of [4, Theorem 3.4.5] yields (2) of the present theorem. The second observation also yields (3). Note that by Proposition 5.9, we see that (H U ) U ∈IC(M,M 1 ) is a measurable field of Hilbert spaces of which (H M 1 U ) U ∈IC(M,M 1 ) forms a measurable field of subspaces. Here we used thatM u is separable.
To prove (4), we make the following observations. First of all, by Remark 6.3, we see that where we use Proposition 5.9 to infer that the direct integral on the right hand side exists. Hence, by Remark 5.11 we see that W E and  3. The support of µ M 1 is given by IR(N c ). Here, IR(N c ) is a subspace of IC(M, M 1 ) as in (6.6). The prove can be done in exactly the same manner as [4,Theorem 3.4.8], see also [7,Proposition 8.6.8]. Note that in the course of the proof of Theorem 7.1 we have already proved that the support of µ M 1 is contained in IR(N c ). Remark 7.4. As pointed out in remark [2,Remark 3.3], the Theorem 7.1 also holds if one assumes thatN c andM c are separable, instead ofN u . Moreover, note that ifM c is separable, then so isN c ⊆M c . IfM is a type I von Neumann algebra, then so isN = P γM P γ . In particular, ifM is type I andM c is separable, then then the result of Theorem 7.1 holds for any closed quantum subgroup of (M, ∆).
8 Example: SU q (1, 1) ext Let (M, ∆) be the quantum group analogue of the normaliser of SU (1, 1) in SL(2, C) as introduced in the operator algebraic framework in [13] and further studied in [10]. In this section we identify the circle as a closed quantum subgroup of (M, ∆). We show that for this pair the map ∆ ♮ defined in (3.3) is not cocommutative. Moreover, the von Neumann subalgebra N of M consisting of bi-invariant elements is not commutative. However, we show how the von Neumann algebras N andN as defined in the previous section can be equipped with a Z 2 -grading. The grading allows us to derive similar results as for (quantum) Gelfand pairs. In particular, we make the Fourier transform explicit and show that it preserves the Z 2 -grading. Moreover, we derive product formulae for little q-Jacobi functions appearing as matrix coefficients of corepresentations which admit invariant vectors. Notation 8.2. In this section we adopt all the notational conventions made in [10]. In particular, in this section we write K instead of H to denote the GNS-space of (M, ∆). For z ∈ C, we denote µ(z) = (z + z −1 )/2. For a set X and x ∈ X, we will write δ x for the function on X that equals 1 in x and 0 elsewhere. It should always be clear from the context what the domain of this function is. Recall in particular that I q = −q N ∪ q Z , where N denotes the natural numbers excluding 0.
For the reader's convenience, we summarize the necessary results on the corepresentation theory of (M, ∆) from [10]. Let W be the multiplicative unitary of (M, ∆) and recall [10] that we have a direct integral decomposition Here σ d (Ω p ) is the discrete spectrum of the Casimir operator [10, Definition 4.5, Theorem 4.6] restricted to the subspace given in [10,Theorem 5.7]. W p,x is a corepresention that is a direct sum of at most 4 irreducible corepresentations [10, Propositions 5.3 and 5.4]. We will simply write W = ⊕ W p,x d(p, x) for the integral decomposition (8.1).
The corepresentations in the continuous part of the decomposition are called principal series corepresentations, the corepresentations that appear as a direct summand are called the discrete series corepresentations. In addition the complementary series corepresentations W p,x , x ∈ (µ(−q), −1) ∪ (1, µ(q)), are defined by analytic continuation of matrix coefficients, see [10, Section 10.3]. We mention that it remains unproved that these make up all the corepresentations.
Using the notation of [10, Sections 10.2 and 10.3], an orthonormal basis for the corepresentation Hilbert space L p,x of the principal and complementary series W p,x is given by the vectors For the discrete series corepresentation W p,x a subset of the vectors (8.2) gives a basis for the corepresentation space L p,x , see [10,Proposition 5.2]. For our purposes, it is convenient to use the following notational convention.
, so that W p,x is a discrete series corepresentation. We denote e ε,η m (p, x), ε, η ∈ {−, +}, m ∈ Z for the zero vector in case e ε,η m (p, x) is not in one of the sets defined in cases 1-3 of [10, Proposition 5.2]. In particular, the non-zero vectors of (8.2) form an orthonormal basis of L p,x .
We remind the reader that the direct integrals over of the vectors f ε,η m (p, x) are vectors in K. Recall that for x ∈ [−1, 1] the actions of the (unbounded) generators of the dual quantum group, see [10, equation (92)], are given by where θ is such that x = µ(e iθ ). Similar expressions can be obtained for the discrete and complementary series corepresentations from the expressions in [10, Lemma 10.1 and Section 10.3].
Remark 8.4. The corepresentations appearing in (8.1) are not mutually inequivalent. We give a complete list of equivalences in Proposition 8.11.
Remark 8.5. For every p ∈ q Z , x ∈ µ(−q 2Z+1 p∪q 2Z+1 p), one can define a corepresentation W p,x by defining the action of the generators ofM by means of [10, Lemma 10.1]. Since the actions of the generators of W pr,x are equal for any r ∈ q 4Z , these corepresentations are all equivalent. Using [10, Proposition 5.2] one can check that every such corepresentation is equivalent to at least one corepresentation in the decomposition (8.1). Since every discrete series corepresentation is infinite dimensional, it occurs infinitely many times in the decomposition (8.1) by the Plancherel theorem [4, Theorem 3.4.1], or see Proposition 8.11 below for a direct proof. Therefore, we see that We will also need the following expressions for the matrix coefficients. By [10, Lemma 10.9 and Section 10.3], for p ∈ q Z , x ∈ (µ(−q), µ(q)), We refer to [10] for the precise definitions of the function C(·) in terms of basic hypergeometric series. (8.4) also holds for the discrete series corepresentations.

The diagonal subgroup
Let T denote the circle group and let (L ∞ (T), ∆ T ) be the usual locally compact quantum group associated with T with dual quantum group (L ∞ (Z), ∆ Z ). We identify (L ∞ (T), ∆ T ) as a closed quantum subgroup of (M, ∆). Recall from [10,Definition 4.3] that the spectrum of the operator K equals 0 ∪ q 1 2 Z .
Note thatπ preserves the comultiplication, since Therefore,π identifies (T, ∆ T ) as a closed quantum subgroup of (M, ∆). Furthermore,π induces a morphism π between the universal quantum groups (M u , ∆ u ) and (C(T), ∆ T ), where here with slight abuse of notation ∆ T is restricted to a map C(T) → C(T × T).

Spherical corepresentations
We compute the actions γ and β of left and right translation and determine which of the corepresentations found in [10] admit a L ∞ (T)-invariant vector.
Proof . Note thatπ has a direct integral decompositionπ = ⊕π p,x d(p, x). Hereπ p,x : where the action of K on the representation space is given by (8.3) and similarly for the discrete series corepresentations.
By the definition of β, see (3.1), and the decomposition (8.1), we see that By (8.3) and the definition ofπ, for almost all pairs (p, x) in the decomposition (8.1), This proves (8.5) for the discrete series corepresentations. It follows from (8.4) and the fact that the function C given there is analytic on a neighbourhood of (µ(−q), µ(q)), see [10,Section 10.3], that for every p ∈ q Z , extends to an analytic function on a neighbourhood of (µ(−q), µ(q)). Here, we use the fact that β is normal and the fact that a function is σ-weak analytic if and only if it is analytic with respect to the norm [20,Result 1.2]. Similarly, it follows that for p ∈ q Z , (8.9) extends to an analytic function on a neighbourhood of (µ(−q), µ(q)). Note that for all p ∈ q Z , x ∈ (µ(−q), µ(q)), 23 ) . Now, (8.7) yields that (8.8) and (8.9) agree on a dense subset of [−1, 1]. Since (8.8) and (8.9) have analytic extensions, they are equal for every x ∈ (µ(−q), µ(q)). The proof of (8.6) is similar.
Remark 8.8. Note that in the preceding proof, we cannot apply [29,Theorem IV.8.25] directly to the mapπ on the von Neumann algebraic level since L ∞ (T) is not separable. Therefore, we have defined the representationsπ p,x explicitly.
Recall that we defined L ∞ (T)-invariant vectors in Definition 5.1.
In Corollary 8.9 we determined the corepresentations appearing in the decomposition that admit a L ∞ (T)-invariant vector. These corepresentations are not mutually inequivalent. Here, we give a list of the equivalences. We only consider the spherical corepresentations and consider the principal, discrete as well as the complementary series. Only the principal and discrete series are important to determine the spaces N ,N and the spherical Fourier transform, see Theorem 7.1. Nevertheless, the complementary series is still important, since the product formulae we derive later still hold for the spherical matrix elements of the complementary series.
Motivated by [10,Lemma 10.11], we introduce the following basis vectors. For p ∈ q 2Z , x ∈ (µ(−q), µ(q)), set For every j ∈ {1, 2}, p ∈ q 2Z , x ∈ (µ(−q), µ(q)), the vectors g j,σ m (p, x), σ ∈ {+, −}, m ∈ Z form an orthonormal basis for L j p,x , the corepresentation space of one of the summands of W p,x , see [10, equation (95)]. For p ∈ q 2Z , x ∈ σ p (Ω d ), set Recall that we made the convention that f ε,η m (p, x) = 0 in case f ε,η m (p, x) is not in the basis given in [10,Proposition 5.2]. Hence, for x ∈ σ d (Ω p ), we see from [10,Proposition 5.2] that the vectors defined in (8.10) are dependent and any of them is equal to a vector of the form f ε,η m (p, x) for some ε, η ∈ {+, −} modulo a phase factor. Now, we determine which of the discrete, principal and complementary series corepresentations are equivalent. By considering the action of the Casimir operator as in the proof of [2,Proposition B.1], it is clear that any two corepresentations that fall within a different series are inequivalent. We restrict ourselves to the spherical corepresentations.
Proof . The proposition follows from a careful comparison of the action of the generators, see In case x = x ′ , an intertwiner must send g j,± m (p, x) to a non-zero scalar multiple of g j ′ ,± m (p ′ , x) as follows by considering the actions of K and E. Writing out the actions of U +− 0 and U −+ 0 one sees that there exists such an intertwiner only in the following two cases: (ii) p/p ′ ∈ q 2+4Z , j = j ′ , in which case it sends g j,± m (p, x) to ±g j ′ ,± m (p ′ , x).
Similarly, in case x = −x ′ , an intertwiner must send g j,± m (p, x) to g j ′ ,∓ m (p ′ , −x) as follows from the actions of K and E. The actions of U +− 0 and U −+ 0 show that this is only possible if (i) p/p ′ ∈ q 4Z , j = j ′ for which it sends g j,± m (p, x) to ±g j,∓ m (p ′ , −x); (ii) p/p ′ ∈ q 2+4Z , j = j ′ , in which case it sends g j,± m (p, x) to g j ′ ,± m (p ′ , −x).
This proves (1), the other cases follows similarly.
Summarizing Corollary 8.9 and Proposition 8.11, we find that IC(M, M 1 ), the space of equivalence classes of irreducible spherical corepresentations is partly given by Here, we identify the points (0, 1, 1), (0, 1, 2), (0, 2, 1), (0, 2, 2) with the respective irreducible corepresentations W The points x ∈ µ(q 2N+1 ) corresponds to W 1,x , see [10,Proposition 5.2]. We emphasize that it is not known if the corepresentations described in [10] are all the corepresentations, therefore we do not know if this description completely describes IC(M, M 1 ). For the von Neumann algebras N andN as well as the spherical Fourier transform, only the principal and discrete (spherical) series matter. These are fully identified within (8.11). For completeness, we illustrate the right hand side of (8.11) by means of Fig. 1.

I II III
· · · · · · · . . . The von Neumann algebra N The next step is to make N andN more explicit and meanwhile define a grading on these spaces. Therefore, we will first find an alternative formula for the mapping T β T γ which is convenient for computations. This is also going to play a role when we derive product formulae. Recall that the spectrum of K is given by q Definition 8.12. For m ∈ Z, p, t ∈ −q Z ∪ q Z , let P m,p,t be the orthogonal rank one projection of K onto the space spanned by the vector f m,p,t .
Here we define f m,p,t to be the zero vector if either p ∈ I q or t ∈ I q . Define mappings T + and T − defined by where the sum converges in the strong topology.
Proof . For a normal functional ω = i∈I ω ξ i ,η i ∈ M * , with i∈I ξ i 2 , i∈I η i 2 < ∞, the linear map M → C : x → ωT ± (x) = i∈I m∈Z,p,t∈Iq xP m,p,t ξ i , P m,±p,t η i , is normal. Hence, the maps T ± are normal. Moreover, for x ∈ M k,l , using (8.4), For x ∈ M k,l we see by using (8.5) and (8.6) that T β T γ (x) also equals the right hand side of (8.12). Since also T β T γ is normal, this proves that T + + T − = T β T γ .
Definition 8.14. We identify N with the von Neumann algebra acting on L 2 (I q ) being generated by L ∞ (I q ) and u 0 . We split N as a direct sum of vector spaces This turns N into a Z 2 -graded algebra.
We find that ϕ ♮ , i.e. the restriction of the Haar weight ϕ [13, Definition 4.1] to N , equals the measure given by: For f ∈ m + ϕ ♮ we find that u 0 f ∈ m ϕ ♮ and it follows by [13,Definition 4.1] that ϕ ♮ (u 0 f ) = 0, so that Λ(n ϕ ∩ N + ) and Λ(n ϕ ∩ N − ) are orthogonal spaces. The discussion so far allows us to make the following identifications.

The von Neumann algebraN
We now turn our attention to the von Neumann algebraN as defined in Section 3. Considered as a subalgebra ofM , it inherits the Z 2 -grading defined in [ The isomorphism is determined by the map Under this isomorphismN + corresponds to the direct integrals over matrices whose entries vanish off the diagonal.N − corresponds to the direct integrals over matrices with values vanishing on the diagonal.
Proof . It follows from Corollary 8.9 that the map defined in (8.14) indeed maps into the proper matrix algebras. Let P ∈M ∩M ′ be the projection as in the proof of [ Since P γ ∈M , it has a direct integral decomposition as in (8.15), i.e.
Remark 8.18. The von Neumann algebraM 0 in the previous proof is isomorphic to 4 copies of C as follows from [10,Proposition 10.13]. Since this does not matter for the integral decomposition (8.13), and to avoid some redundant extra notation we have not treated the point x = 0 separately.
We claim thatφ ♮ , i.e. the restriction of the dual left Haar weight toN , is a trace. Indeed, it follows from [2, Section 5] and [4, Proposition 3.5.5] that for any p ∈ q Z , x ∈ [−1, 1] ∪ σ d (Ω p ), there is a constant c(p, x) such that Here D = ⊕ D p,x d(p, x), with short hand notation D p,x = D Wp,x , is the Duflo-Moore operator arising from the Plancherel theorem, see [4,Theorem 3.4.1] and also Theorem 7.1. Since the eigenvalues of D p,x are independent of the signs ε, η, we see that is the Radon-Nikodym derivative ofφ ♮ with respect to a trace, see the proof of [4, Theorem 3.4.5], we see thatφ ♮ is a trace. Therefore, under the identification (8.13), we see that where d is a scalar valued function, which we leave undetermined.
Remark 8. 19. It follows from [4, Theorem 3.4.1 (6)] that the function c(1, x) depends on the Plancherel measure. In case the Plancherel measure is choosen as in (8.1), one has d(x) = c(1, x) −2 . For the discrete series these constants are computed in [4,Section 3.5]. For the principal series, the exact values cannot be found in the literature. We will not derive them here, since this is beyond the scope of our example.
We identify the GNS-space of ϕ ♮ using the isomorphism (8.13). That is, the GNS-space is given by where the direct integral and direct sums are taken as Hilbert spaces (as opposed to the direct integrals and sums of von Neumann algebras given in (8.13)), where the inner product comes from the traces d(x)(Tr M 2 (C) ⊕ Tr M 2 (C) ) for the integral part and d(x)Tr C for the direct summands.

The spherical Fourier transform
Here, we determine the spherical Fourier transform, i.e. we determine the map Q L ∞ (T) 0 for Theorem 7.1. In particular, we are interested in the kernels of the integral transformations that appear in this transform, since we will need them later. Using the identifications of the GNSspaces for ϕ ♮ andφ ♮ with L 2 (N ) and L 2 (N ), we may consider Q L ∞ (T) 0 as a map from L 2 (N ) to L 2 (N ) and use the short hand notation F 2 : L 2 (N ) → L 2 (N ) for this map.
If f ∈ L 1 (I q ) ∩ L ∞ (I q ) ⊆ N + , then there is a functional f · ϕ ♮ ∈ N * given by (f · ϕ ♮ )(x) = ϕ ♮ (xf ), x ∈ N . Then, ξ(f · ϕ ♮ ) = Λ(f ) by definition of ξ, see Section 2. Thus, for such f , we find by Theorem 7.1 that, Here, the right hand side indeed is an element of L 2 (N ) by Corollary 8.9. Note that we choose the inner product on L 2 (N ) to be the direct integral over the traces d(x)(Tr M 2 (C) ⊕ Tr M 2 (C) ) for the integral part and d(x)Tr C for the direct summands. Hence, the Duflo-Moore operators which are direct integrals of scalar multiples of the identity are contained in the inner product by means of the function d. Hence, they do not appear in (8.17). Similarly, for f ∈ L 1 (I q )∩ L ∞ (I q ), so that f u 0 ∈ N − , We see from (8.17) that the spherical Fourier transformation is in fact a combination of integral transformations with the spherical matrix elements of the corepresentations W 1,x in its kernel. The next step is to make these kernels explicit.