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,C) to...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2011 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2011
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147385 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859636045506674688 |
|---|---|
| author | Caspers, M. |
| author_facet | Caspers, M. |
| citation_txt | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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,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 Z₂-grading, we obtain product formulae for little q-Jacobi functions.
|
| first_indexed | 2025-12-07T13:15:48Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 087, 39 pages
Spherical Fourier Transforms
on Locally Compact Quantum Gelfand Pairs?
Martijn CASPERS
Radboud Universiteit Nijmegen, IMAPP, Heyendaalseweg 135,
6525 AJ Nijmegen, The Netherlands
E-mail: caspers@math.ru.nl
URL: http://www.math.ru.nl/~caspers/
Received April 14, 2011, in final form August 30, 2011; Published online September 06, 2011
http://dx.doi.org/10.3842/SIGMA.2011.087
Abstract. 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,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 Z2-grading, we obtain product
formulae for little q-Jacobi functions.
Key words: locally compact quantum groups; Plancherel theorem; Fourier transform; sphe-
rical functions
2010 Mathematics Subject Classification: 16T99; 43A90
1 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-K-
invariant L1-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 (SO0(1, n), SO(n)), where
SO0(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-)al-
gebraic 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.
?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe-
cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSF.html
mailto:caspers@math.ru.nl
http://www.math.ru.nl/~caspers/
http://dx.doi.org/10.3842/SIGMA.2011.087
http://www.emis.de/journals/SIGMA/OPSF.html
2 M. Caspers
For quantum groups, many deformations of classical Gelfand pairs do indeed form a quantum
Gelfand pair that moreover is strict. As a compact example, (SUq(n), Uq(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 1ϕ1 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 SUq(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 SUq(2) and get so called (σ, τ)-spherical elements, see [15] for this
particular example. See also [14] for a similar study of SUq(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 be-
tween 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 L2-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-deforma-
tion 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 SUq(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 SUq(1, 1)ext
with its diagonal subgroup. Nevertheless, the pair exhibits properties reminiscent of classical
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 3
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 SUq(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 and N̂ as
well as the C∗-algebras N̂c and N̂u. 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 corepre-
sentations of the quantum group itself. This will form the essential bridge between [4, Theo-
rem 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 SUq(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.
2 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 W23 for 1⊗W and W13 = (Σ⊗ 1)W23(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 denote ω̄ for the normal functional defined
by ω̄(x) = ω(x∗). For ω ∈M∗ and x, y ∈M , we denote xω, ωy, xωy for the functionals defined
by (xω)(z) = ω(zx), (ωy)(z) = ω(yz), (xωy)(z) = ω(yzx), where z ∈M .
4 M. Caspers
Let φ be a weight on M . Let nφ = {x ∈ M | φ(x∗x) < ∞}, mφ = n∗φnφ. Let σφ be the
modular automorphism group of φ. We denote Tφ for the Tomita algebra defined by
Tφ =
{
x ∈M | x is analytic w.r.t. σφ and ∀ z ∈ C : σφz (x) ∈ nφ ∩ n∗φ
}
.
For x, y ∈ Tφ, 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].
Definition 2.1. A locally compact quantum group (M,∆) consists of the following data:
1. A von Neumann algebra M ;
2. A unital, normal ∗-homomorphism ∆ : M →M⊗M satisfying the coassociativity relation
(∆⊗ ι)∆ = (ι⊗∆)∆;
3. Two normal, semi-finite, faithful weights ϕ, ψ on M so that
ϕ ((ω ⊗ ι)∆(x)) = ϕ(x)ω(1), ∀ω ∈M+
∗ , ∀x ∈ m+
ϕ (left invariance);
ψ ((ι⊗ ω)∆(x)) = ψ(x)ω(1), ∀ω ∈M+
∗ , ∀x ∈ m+
ψ (right invariance).
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 ψσt = ν−tψ. By applying [34], we see
that there is a positive, self-adjoint operator δ, called the modular element, such that ψ = ϕδ,
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
W ∗ (Λ(a)⊗ Λ(b)) = (Λ⊗ Λ) (∆(b)(a⊗ 1)) , a, b ∈ nϕ.
W is known as the multiplicative unitary. It satisfies the pentagon equation W12W13W23 =
W23W12 in B(H⊗H⊗H). Here we use the usual leg numbering notation. Moreover, W imple-
ments the comultiplication, i.e. ∆(x) = W ∗(1⊗ x)W and W ∈M ⊗B(H).
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 ) 7→ (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
S = Rτ−i/2, R2 = ι, τtR = Rτt, ϕτt = ν−tϕ, ∀ t ∈ R.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 5
R is called the unitary antipode and τ is called the scaling group. Moreover,
∆R = ΣM,M (R⊗R)∆, ψ = ϕR, (2.1)
where ΣM,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 itΛ(x) = ν
t
2 Λ(τt(x)), t ∈ R, x ∈ nϕ. We use the notation
M ]
∗ =
{
ω ∈M∗ | ∃ θ ∈M∗ : (θ ⊗ ι)(W ) = (ω ⊗ ι)(W )∗
}
.
For ω ∈ M ]
∗, ω
∗ is defined by (ω∗ ⊗ ι)(W ) = (ω ⊗ ι)(W )∗. 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.
The dual quantum group
In [21, 22], it is proved that there exists a dual locally compact quantum group (M̂, ∆̂), so that
(
ˆ̂
M,
ˆ̂
∆) = (M,∆). The dual left and right Haar weight are denoted by ϕ̂ and ψ̂. Similarly, all
other dual objects will be denoted by a hat, i.e. ∇̂, Ĵ , δ̂, σ̂t, Ŵ , . . .. P is self-dual, i.e. P̂ = P . By
construction,
M̂ =
{
(ω ⊗ ι)(W ) | ω ∈M∗
}σ-strong-∗
.
Furthermore, Ŵ = ΣW ∗Σ, where Σ denotes the flip on H⊗H. This implies that W ∈M ⊗ M̂
and
M =
{
(ι⊗ ω)(W ) | ω ∈ M̂∗
}σ-strong-∗
.
For ω ∈ M∗, we use the standard notation λ(ω) = (ω ⊗ ι)(W ) ∈ M̂ . Then λ : M ]
∗ → M̂ is
a representation.
We denote I for the set of ω ∈ M∗, such that Λ(x) 7→ ω(x∗), x ∈ nϕ, extends to a bounded
functional on H. By the Riesz theorem, for every ω ∈ I, there is a unique vector denoted by
ξ(ω) ∈ H such that ω(x∗) = 〈ξ(ω),Λ(x)〉, x ∈ nϕ. The dual left Haar weight ϕ̂ is defined to be
the unique normal, semi-finite, faithful weight on M̂ , with GNS-construction (H, ι, Λ̂) such that
λ(I) is a σ-strong-∗/norm core for Λ̂ and Λ̂(λ(ω)) = ξ(ω), ω ∈ I. By definition, this means that
{(λ(ω), ξ(ω)) | ω ∈ I} is dense in the graph of Λ̂ with respect to the product of the σ-strong-∗
topology on M and the norm topology on H.
Corepresentations
A (unitary) corepresentation is a unitary operator U ∈ M ⊗ B(HU ) that satisfies the relation
(∆ ⊗ ι)(U) = U13U23. 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 GNS-
space H. Two corepresentations U1, U2 are equivalent if there is a unitary T : HU1 → HU2 such
that T (ω⊗ ι)(U1)v = (ω⊗ ι)(U2)Tv for every ω ∈M∗, v ∈ HU1 . We denote IC(M) for the set of
equivalence classes of all unitary corepresentations. For any corepresentation U ∈ M ⊗ B(HU )
and ω ∈ B(HU )∗, (ι⊗ ω)(U) ∈ Dom(S) and
S(ι⊗ ω)(U) = (ι⊗ ω)(U∗).
6 M. Caspers
Reduced quantum groups
To every locally compact quantum group (M,∆) one can associate a reduced C∗-algebraic quan-
tum group. We define Mc to be the norm closure of {(ι⊗ω)(W ) | ω ∈ M̂∗}, which is a C∗-algebra.
We restrict ∆, ϕ and ψ to Mc and denote the respective restrictions by ∆c, ϕc, ψc. In fact,
∆c should be considered as a map into the multiplier algebra of the minimal tensor product
of Mc with itself. The GNS-construction representation (H,Λ, π) then restricts to a GNS-
representation of the C∗-algebraic weight ϕc, which is denoted by (H,Λc, πc). It is proven in [21]
that (Mc,∆c) forms a reduced C∗-algebraic quantum group. Similarly, one defines the reduced
dual C∗-algebraic quantum group (M̂c, ∆̂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. Let M̂u
be the completion of M ]
∗ with respect to ‖ · ‖u. We let λu : M ]
∗ → M̂u denote the canonical
embedding. M̂u carries the following universal property: if π is a representation ofM ]
∗, then there
is a unique representation ρ of M̂u such that π = ρλu. In particular, from the representation λ
we get a surjective map ϑ̂ : M̂u → M̂c. We define a universal weight on M̂u by setting ϕ̂u = ϕ̂cϑ̂,
and ψ̂u = ψ̂cϑ̂. The GNS-representation of ϕ̂u is given by (H, Λ̂u = Λ̂cϑ̂, π̂u = π̂cϑ̂). If U ∈
M⊗B(HU ) is a corepresentation of M , then the map M ]
∗ → B(HU ) : ω 7→ (ω⊗ι)(U) determines
a representation of M̂u, which we denote by πU . In fact, it is shown in [18] that this establishes
a 1-1 correspondence between corepresentations of M and non-degenerate representations of M̂u.
For completeness, we mention that M̂u can be equipped with a comultiplication ∆̂u, which
is a map from M̂u to the multiplier algebra of the minimal tensor product of M̂u with itself,
such that (M̂u, ∆̂u) is a universal C∗-algebraic quantum group in the sense of [18]. Similarly,
one defines Mu,∆u, ϑ, ϕu, ψu, . . ..
3 Spherical Fourier transforms
Let (G,K) be a pair consisting of a locally compact group G and a compact subgroup K. If
L1(K\G/K), the bi-K-invariant L1-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, H1. Denote ∆ for the
comultiplication of H, denote ϕ1 for the Haar functional on H1. Suppose that there exists
a surjective morphism π : H → H1, so that H1 is identified as a quantum subgroup of H. Now
consider the left and right coactions ∆l = (π ⊗ ι)∆,∆r = (ι⊗ π)∆. Define
H1\H =
{
h ∈ H | ∆l(h) = 1⊗ h
}
, H/H1 =
{
h ∈ H | ∆r(h) = 1⊗ h
}
and set H1\H/H1 = (H1\H) ∩ (H/H1). Classically, H1\H/H1 corresponds to the algebra of
bi-K-invariant elements. Set
∆̃ = (ι⊗ ϕ1π ⊗ ι)(ι⊗∆)∆.
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.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 7
Definition 3.1. Let (H,H1) be as above. (H,H1) is called a Gelfand pair if ∆̃ is cocommutative,
i.e. ∆̃ = ΣH,H∆̃. The pair is called a strict Gelfand pair if moreover H1\H/H1 is commutative.
Here ΣH,H denotes the flip.
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 sphe-
rical 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 SUq(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 (M1,∆1) which we assume to be the compact. Recall
[37, Definition 2.9] that this means that we have a surjective ∗-homomorphism π : Mu → (M1)u
on the level of universal C∗-algebras and the induced dual ∗-homomorphism π̂ : (M̂1)u → M̂u
lifts to a map on the level of von Neumann algebras π̂ : M̂1 → M̂ , which with slight abuse of
notation is denoted by π̂ again. When we encounter π̂ in this paper, we always mean the von
Neumann algebraic map.
Note π and π̂ are in principal also used for the GNS-representations of M and M̂ . However, we
omit the maps most of the time, since M and M̂ 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.
We use ΣM1,M : M1⊗M →M⊗M1 to denote the flip. The objects associated with (M1,∆1)
will be equipped with a subscript 1, i.e. S1, R1, τ1, ν1, ϕ1, . . ..
Homogeneous spaces
Due to [36, Proposition 3.1], there are canonical right coactions of (M1,∆1) on M , denoted by
β, γ : M →M ⊗M1, which are normal ∗-homomorphisms uniquely determined by
(β ⊗ ι)(W ) = W13 ((ι⊗ π̂)(W1))23 , (γ ⊗ ι)(W ) = ((R1 ⊗ π̂)(W1))23W13. (3.1)
We have the relation γ = (R ⊗ ι)βR. The map β corresponds classically to right translation,
whereas γ corresponds to left translation.
Remark 3.3. In [36, Section 3], the roles of (M1,∆1) and (M̂1, ∆̂1) are interchanged. Using
our conventions for the roles of (M1,∆1) and (M̂1, ∆̂1), recall the left action µ of (M1,∆1) on M
and the left action θ of (M1,∆
op) on M from [36, Proposition 3.1]. By this proposition, β equals
the right coaction ΣM1,Mθ. γ equals the right coaction ΣM1,M (R1 ⊗ ι)µ.
Lemma 3.4. As maps M →M ⊗M ⊗M1, we have an equality
(ι⊗ ΣM1,M )(β ⊗ ι)∆ = (ι⊗ ι⊗R1)(ι⊗ γ)∆. (3.2)
8 M. Caspers
Proof. For the left hand side, using the pentagon equation and [36, Proposition 3.1],
(β ⊗ ι⊗ ι)(∆⊗ ι)W = (β ⊗ ι⊗ ι)W13W23 = W14((ι⊗ π̂)(W1))24W34.
For the right hand side, using again [36, Proposition 3.1],
(ι⊗ ι⊗R1 ⊗ ι)(ι⊗ γ ⊗ ι)(∆⊗ ι)W = (ι⊗ ι⊗R1 ⊗ ι)(ι⊗ γ ⊗ ι)W13W23
= (ι⊗ ι⊗R1 ⊗ ι)W14((R1 ⊗ ι)(ι⊗ π̂)(W1))34W24 = W14((ι⊗ π̂)(W1))34W24.
The lemma follows by the fact that the elements {(ι ⊗ ω)(W ) | ω ∈ M̂∗} are σ-strong-∗ dense
in M . �
Definition 3.5. We denote Mβ for the fixed point algebra {x ∈M | β(x) = x⊗ 1}. Similarly,
Mγ denotes the fixed point algebra of γ. By definition of γ we find Mγ = R(Mβ). We define
N = Mβ ∩Mγ .
Note that Mβ, Mγ and N are von Neumann algebras. Furthermore, ∆(Mβ) ⊆ M ⊗Mβ.
Also, by Mγ = R(Mβ), (2.1) and (3.2) it follows that ∆(Mγ) ⊆Mγ ⊗M .
We recall from [35] that we have normal, faithful operator valued weights,
Tβ : M+ → (Mβ)+ : x 7→ (ι⊗ ϕ1)β(x); Tγ : M+ → (Mγ)+ : x 7→ (ι⊗ ϕ1)γ(x).
Since (M1,∆1) is compact, Tβ and Tγ are finite. We extend the domains of Tβ and Tγ to M in
the usual way. We denote the extensions again by Tβ and Tγ . The composition of Tβ and Tγ
forms a well-defined map on M . Note that Tβ(x∗) = Tβ(x)∗ and Tγ(x∗) = Tγ(x)∗, where x ∈M .
Remark 3.6. The spaces Mγ , Mβ where already introduced in [33] as homogeneous spaces.
They also fall within the definition of a homogeneous space as introduced by Kasprzak [12,
Remark 3.3]. Moreover, we stretch that Tβ and Tγ are conditional expectation values, which
properties have been studied in the related papers [28] and [32].
Lemma 3.7. Tγ : M →Mγ and Tβ : M →Mβ satisfy the following properties:
1. TβTγ = TγTβ;
2. ∆Tβ = (ι⊗ Tβ)∆ and ∆Tγ = (Tγ ⊗ ι)∆;
3. (ι⊗ Tγ)∆ = (Tβ ⊗ ι)∆;
4. TγS ⊆ STβ and TβS ⊆ STγ.
Proof. (1) This follows from the fact (ι⊗ΣM1,M1)(γ⊗ ι)β = (β⊗ ι)γ, which can be established
as in the proof of Lemma 3.4.
(2) We prove that ∆Tγ = (Tγ ⊗ ι)∆, the other equation can be proved similarly using
β = (R⊗ ι)γR and (2.1). We find:
(∆Tγ ⊗ ι)(R⊗ ι)(W ) = (ι⊗ ι⊗ ϕ1 ⊗ ι)(∆⊗ ι⊗ ι)(R⊗ ι⊗ ι)(β ⊗ ι)(W )
= (ΣM,M ⊗ ι)(R⊗R⊗ ι)(∆⊗ ι)(W (1⊗ π̂((ϕ1 ⊗ ι)(W1))))
= (R⊗R⊗ ι)W23W13(1⊗ 1⊗ π̂((ϕ1 ⊗ ι)(W1)))
= (ι⊗ ϕ1 ⊗ ι⊗ ι)(R⊗ ι⊗ ι⊗ ι)(β ⊗R⊗ ι)W23W13
= (Tγ ⊗ ι⊗ ι)(ΣM,M ⊗ ι)(R⊗R⊗ ι)W13W23
= (Tγ ⊗ ι⊗ ι)(∆⊗ ι)(R⊗ ι)(W ).
Now, the equation follows by taking slices of the second leg of W .
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 9
(3) Since (M1,∆1) is compact, it is unimodular and hence ϕ1 = ϕ1R1. Using Lemma 3.4 we
find:
(ι⊗ Tγ)∆ = (ι⊗ ι⊗ ϕ1)(ι⊗ γ)∆ = (ι⊗ ι⊗ ϕ1R1)(ι⊗ γ)∆
= (ι⊗ ι⊗ ϕ1)Σ23(β ⊗ ι)∆ = (Tβ ⊗ ι)∆.
(4) It follows from [22, Proposition 2.1, Corollary 2.2] that (τt⊗ ι)(W ) = (ι⊗ τ̂−t)(W ), t ∈ R.
Therefore,
(βτt ⊗ ι)(W ) = (β ⊗ τ̂−t)(W ) = (ι⊗ ι⊗ τ̂−t)(W13((ι⊗ π̂)(W1))23).
By [21, Proposition 5.45], we know that π̂(τ̂1)t = τ̂tπ̂. Here (τ̂1)t denotes the scaling group of
(M̂1, ∆̂1). Continuing the equation, we find
(βτt ⊗ ι)(W ) = (ι⊗ τ̂−t)(W )13((ι⊗ π̂(τ̂1)−t)(W1))23
= (τt ⊗ ι)(W )13((τ1)t ⊗ π̂)(W1))23 = (τt ⊗ (τ1)t ⊗ ι)(β ⊗ ι)(W ).
We have ϕ1(τ1)t = ϕ1, since (M1,∆1) is compact. Hence, Tβτt = τtTβ. With a similar com-
putation, involving the relation (R ⊗ R̂)(W ) = W ∗, see [22, Proposition 2.1, Corollary 2.2], we
find TβR = RTγ . The proposition follows, since S = Rτ−i/2. �
Definition 3.8. For x ∈ N , we define
∆\(x) = (ι⊗ Tγ)∆(x) = (Tβ ⊗ ι)∆(x), (3.3)
see Lemma 3.7 (3). This is the von Neumann algebraic version of [40, equation (4)].
Recall that ∆(N) ⊆ Mγ ⊗Mβ, so that ∆\(N) ⊆ N ⊗ N . Moreover, using (1)–(3) of the
previous lemma, ∆\ is coassociative, i.e.
(ι⊗∆\)∆\ = (ι⊗ ι⊗ Tγ)(ι⊗∆)(ι⊗ Tγ)∆ = (ι⊗ TβTγ ⊗ ι)(ι⊗∆)∆
= (ι⊗ TγTβ ⊗ ι)(∆⊗ ι)∆ = (∆\ ⊗ ι)∆\. (3.4)
Note that ∆\ is unital, but generally not multiplicative.
Definition 3.9. We define a convolution product ∗\ on N∗,
ω1 ∗\ ω2 = (ω1 ⊗ ω2)∆\, ω1, ω2 ∈ N∗.
This convolution product is associative, since by (3.4) ∆\ is coassociative.
Definition 3.10. If ω ∈ N∗, then we define
ω̃ = ωTγTβ = ωTβTγ ∈M∗.
For ω ∈M∗, we put ω̃ = (ω|N )∼.
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].
10 M. Caspers
Proposition 3.12. The map N∗ 3 ω 7→ ω̃ defines a bijective, norm preserving correspondence
between N∗ and the functionals θ ∈M∗ that satisfy the invariance properties:
1. (θ ⊗ ι)β(x) = θ(x)1M1 for all x ∈M ;
2. (θ ⊗ ι)γ(x) = θ(x)1M1 for all x ∈M .
Moreover, for ω1, ω2 ∈ N∗ we have
(
ω1 ∗\ ω2
)∼
= ω̃1 ∗ ω̃2.
Proof. Using right invariance of ϕ1, for x ∈M ,
(Tβ ⊗ ι)β(x) = (ι⊗ ϕ1 ⊗ ι)(β ⊗ ι)β(x) = (ι⊗ ϕ1 ⊗ ι)(ι⊗∆1)β(x)
= (ι⊗ ϕ1)β(x)⊗ 1M1 = Tβ(x)⊗ 1M1 .
Similarly, the right invariance of ϕ1 implies (Tγ ⊗ ι)γ(x) = Tγ(x) ⊗ 1M1 , x ∈ M . Using (1) of
Lemma 3.7 one easily verifies that for ω ∈ N∗, ω̃ satisfies the invariance properties (1) and (2).
Let ω ∈ N∗. For x ∈ N we have ω(x) = ω̃(x), so that N∗ 3 ω 7→ ω̃ is injective. If θ ∈ M∗
satisfies (1), then for x ∈M ,
θTβ(x) = θ(ι⊗ ϕ1)β(x) = ϕ1(θ ⊗ ι)β(x) = θ(x).
Similarly, if θ ∈M∗ satisfies (2), then θTγ(x) = θ(x). We find that θ = (θ|N )∼ if θ satisfies (1)
and (2). So N∗ 3 ω 7→ ω̃ ranges over the normal functionals on M that satisfy the invariance
properties (1) and (2).
Using the left invariance of ϕ1, it is a straightforward check that TγTγ = Tγ . Then, using
(1)–(3) of Lemma 3.7, we find:
(ω1 ∗\ ω2)∼ = (ω1 ⊗ ω2)(ι⊗ Tγ)∆TγTβ = (ω1 ⊗ ω2)(Tγ ⊗ TγTβ)∆
= (ω1 ⊗ ω2)(Tγ ⊗ T 2
γTβ)∆ = (ω1 ⊗ ω2)(TγTβ ⊗ TγTβ)∆ = ω̃1 ∗ ω̃2. �
Proposition 3.13. For ω ∈M ]
∗, we have ω̃ ∈M ]
∗ and (ω̃)∗ = (ω∗)∼.
Proof. Using (4) of Lemma 3.7, for x ∈ Dom(S), we find
ω̃(S(x)) = ω(TβTγ(S(x)∗)) = ω((TβTγ(S(x)))∗) = ω(S(TβTγ(x))∗) = (ω∗)∼(x).
So (ω∗)∼ ∈ M∗ has the property ((ω∗)∼ ⊗ ι)(W ) = (ω̃ ⊗ ι)(W )∗. This proves that ω̃ ∈ M ]
∗ and
ω̃∗ = (ω∗)∼. �
The following proposition is proved in [36, Proposition 3.1].
Proposition 3.14. If x ∈ nϕ, then Tγ(x) ∈ nϕ and Λ(Tγ(x)) = π̂((ϕ1 ⊗ ι)(W1))Λ(x).
Proposition 3.14 defines an orthogonal projection π̂((ϕ1⊗ ι)(W1)) for which we simply write
Pγ = π̂((ϕ1 ⊗ ι)(W1)).
Classically, it corresponds to projecting onto the space of functions that are left invariant with
respect to the compact subgroup. Note that Pγ ∈ M̂ and
N = TβTγ(M) =
{
(ι⊗ ωPγv,Pγw)(W ) | v, w ∈ H
}
.
We need a similar result as Proposition 3.14 for Tβ. 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
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 11
definition of a Gelfand pair a bit to handle the example of SUq(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 ∈ Tϕ, we have aϕb ∈ M∗ and hence
TϕϕTϕ 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 ∈ nϕ. By [21, Lemma 8.5], TϕϕTϕ is
included in I.
Lemma 3.15. Let ω ∈ I. There exists a net (ωj)j in TϕϕTϕ such that ωj → ω in norm and
ξ(ωj)→ ξ(ω) in norm.
Proof. For ω ∈ I we define the norm ‖ω‖I = max{‖ω‖, ‖ξ(ω)‖}. We have to prove that
TϕϕTϕ 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],
T 2
ϕ ⊆ L and for a, b ∈ Tϕ, k(ab) = σi(b)ϕa, see also [1, Corollary 2.15]. So k(T 2
ϕ ) = TϕϕTϕ. The
proposition yields that k(T 2
ϕ ) is dense in I. �
Proposition 3.16. Suppose that (M,∆) is unimodular. For x ∈ nϕ, we have Tβ(x) ∈ nϕ. The
map Λ(x) 7→ Λ(Tβ(x)) is bounded and it extends continuously to the projection ĴPγ Ĵ .
Proof. By Proposition 3.14, we see that Λ(Tγ(x∗)) = Λ(Tγ(x)∗), x ∈ nϕ∩n∗ϕ. Denote T for the
closure of the map Λ(x) 7→ Λ(x∗), x ∈ nϕ ∩ n∗ϕ. We see that PγH is an invariant subspace for T .
Since T = J∇1/2, we find that ∇it, t ∈ R commutes with Pγ .
Recall [21, Lemma 8.8, Proposition 8.9] that ∇̂it = P itJδitJ and by Pontrjagin duality
∇it = P̂ itĴ δ̂itĴ . Using δ = 1 and the self-duality P̂ = P , we see that ∇̂it = ∇itĴ δ̂−itĴ . Since δ̂
is affiliated with M̂ and using the previous paragraph, this shows that Pγ∇̂it = ∇̂itPγ . Hence
σ̂t(Pγ) = Pγ .
Now, let a, b ∈ Tϕ̂ and put ω = aϕ̂b ∈ M̂∗. Then,
Tβ ((ι⊗ ω)(W ∗)) = (ι⊗ ω) ((1⊗ Pγ)W ∗) .
Since Pγ ∈ M̂ is invariant under σ̂t, it follows from [30, Chapter VIII, Lemma 2.4 (ii) and
Lemma 2.5] that ωPγ ∈ Î and
ξ̂(ωPγ) = Λ̂(aσ̂−i(b)σ̂−i(Pγ)) = ĴPγ ĴΛ̂(aσ̂−i(b)).
Now, let ω ∈ Î. We prove the proposition for x = λ̂(ω) ∈ nϕ. Let (ωj)j∈J be a net in Tϕ̂ϕ̂Tϕ̂
that converges to ω and such that ξ̂(ωj) converges to ξ̂(ω), c.f. Lemma 3.15. Then, Tβ(λ̂(ωj)) ∈
nϕ and Tβ(λ̂(ωj))→ Tβ(x) in the σ-weak topology. Furthermore, Λ(Tβ(λ̂(ωj))) = ĴPγ Ĵ ξ̂(ωj) is
norm convergent. Since Λ is σ-weak/weak closed, and Dom(Λ) = nϕ, this proves that Tβ(x) ∈ nϕ
and Λ(Tβ(x)) = ĴPγ ĴΛ(x).
Since the elements in λ̂(Î) form a σ-strong-∗/norm core for Λ, this proves the proposition. �
We will write Pβ for the projection ĴPγ Ĵ . In particular Pβ ∈ M̂ ′. Under the assumption
that (M,∆) is unimodular, we see that Pβ projects onto the elements that are right invariant
with respect to the closed quantum subgroup (M1,∆1). Since we will need this interpretation
of Pβ, i.e. Proposition 3.16, we assume unimodularity from now on.
Notation 3.17. From this point we assume that (M,∆) is unimodular, i.e. ϕ = ψ.
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
12 M. Caspers
C∗-algebraic quantum group. These duals can be constructed by means of the multiplicative
unitary W associated with (M,∆). We define
N ]
∗ =
{
ω ∈ N∗ | ∃ θ ∈ N∗ : (ω̃ ⊗ ι)(W )∗ = (θ̃ ⊗ ι)(W )
}
.
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 if ω̃ ∈M ]
∗. Note that N ]
∗ is dense
in N∗. Indeed, the restriction map M∗ → N∗ : ω 7→ ω|N is continuous and the image of the
subset M ]
∗ ⊆ M∗ is contained in N ]
∗ by Proposition 3.13. The inclusion M ]
∗ ⊆ M∗ is dense, see
[22, Lemma 2.5]. Hence N ]
∗ is dense in N∗. Using this together with Propositions 3.12 and 3.13,
we see that
N̂ = {(ω̃ ⊗ ι)(W ) | ω ∈ N∗}
σ-strong-∗
,
is a ∗-subalgebra of M̂ . Since we can conveniently write (ω̃⊗ ι)(W ) = Pγ(ω⊗ ι)(W )Pγ by (3.1),
we see that N̂ is a von Neumann algebra if considered as acting on PγH, so that Pγ is its unit.
In particular N̂ = PγM̂Pγ .
We define N̂c to be the norm closure of the set {(ω̃ ⊗ ι)(W ) | ω ∈ N∗}. Then N̂c is a C∗-
subalgebra of the reduced dual C∗-algebra M̂c.
For ω ∈ N ]
∗, we define
‖ω‖\u = sup
{
‖π(ω)‖ | π a representation of N ]
∗
}
.
Note that this defines a norm since the representation ω 7→ (ω̃ ⊗ ι)(W ) is injective as follows
using the bijective correspondence established in Proposition 3.12. Let N̂u be the completion
of N ]
∗ with respect to ‖ · ‖\u.
Recall the map λ : M ]
∗ → M̂ : ω 7→ (ω ⊗ ι)(W ). We set
λ\ : N ]
∗ → N̂ : ω 7→ (ω̃ ⊗ ι)(W ).
Note that the image of λ\ is contained in N̂c and we will use this implicitly. λu : M ]
∗ → M̂u is
the canonical inclusion and similarly
λ\u : N ]
∗ → N̂u
denotes the canonical inclusion.
Recall that N̂u 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 of N̂u such that π = ρλ\u. By
this universal property, the map N ]
∗ → M̂u : ω 7→ λu(ω̃) extends to a representation
ιu : N̂u → M̂u.
Similarly, the map λ\ : N ]
∗ → N̂c gives rise to a surjective map
ϑ̂\ : N̂u → N̂c.
In particular, ϑ̂ιu = ϑ̂\, where ϑ̂ : M̂u → M̂c was the canonical projection induced by the
representation λ : M ]
∗ → M̂c.
Remark 3.18. Note that we do not claim that ιu : N̂u → M̂u is injective. In fact, this
is generally not true, see the comments made in Remark 6.4 and the paragraph before this
remark.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 13
4 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 weights ϕ̂u, ϕ̂c were defined in Section 2. The weights on the von
Neumann algebras M , M̂ and the C∗-algebras M̂c, M̂u restrict to weights on N , N̂ and N̂c, N̂u
by setting
ϕ\ = ϕ|N , ϕ̂\ = ϕ̂|N̂ and ϕ̂\c = ϕ̂c|N̂c , ϕ̂\u = ϕ̂uιu = ϕ̂cϑ̂ιu = ϕ̂\cϑ̂
\,
respectively. We prove that ϕ\ and ϕ̂\ are normal, semi-finite, faithful weights and ϕ̂\c and ϕ̂\u
are lower semi-continuous, densely-defined, non-zero weights. Here the assumption made in
Notation 3.17 becomes essential.
Proposition 4.1. ϕ\ is a normal, semi-finite, faithful weight on N . Its GNS-representation is
given by (PγPβH,Λ|N∩nϕ , π|N ).
Proof. Trivially, ϕ\ is normal and faithful. Since nϕ is σ-weakly dense in M , Tβ and Tγ are
σ-weakly continuous and N = TγTβ(M), Propositions 3.14 and 3.16 prove that ϕ\ is semi-
finite. It is straightforward to check that (PγPβH,Λ|N∩nϕ , π|N ) satisfies all the properties of
a GNS-representation [30, Section VII.1]. �
Proposition 4.2. For ω ∈ I, we find ω̃ ∈ I and ξ(ω̃) = PβPγξ(ω). The set IN = {ω ∈ N∗ |
ω̃ ∈ I} is dense in N∗. The set {ξ(ω̃) | ω ∈ IN} is a dense subset of PβPγH.
Proof. For x ∈ nϕ, using Propositions 3.14 and 3.16,
Λ(x) 7→ ω(TβTγ(x∗)) = ω((TβTγ(x))∗) = 〈ξ(ω),Λ(TβTγ(x))〉 = 〈PβPγξ(ω),Λ(x)〉.
The first claim now follows by the definitions of I and ξ(·). Moreover, we find IN = {ω|N | ω ∈
I}, so that the second claim follows by [21, Lemma 8.5]. The last claim also follows from [21,
Lemma 8.5]. �
Proposition 4.3. ϕ̂\ is a normal, semi-finite, faithful weight on N̂ . Its GNS-representation is
given by (PγPβH, Λ̂|N̂∩nϕ̂ , π̂|N̂ ).
Proof. By Proposition 4.2, {(ω̃ ⊗ ι)(W ) | ω ∈ IN} ⊆ N̂ is a σ-strong-∗ dense subset of N̂
contained in nϕ̂. This proves that ϕ̂\ is semi-finite. Trivially, ϕ̂\ is normal and faithful.
To prove the claim about the GNS-representation, we only need to show that the image
of Λ̂|N̂∩nϕ̂ is contained in PβPγH and that the inclusion is dense. For ω ∈ IN , we have λ(ω̃) ∈
N̂ ∩ nϕ̂ and Λ̂(λ(ω̃)) = ξ(ω̃) ∈ PβPγH by Proposition 4.2. Now, let x ∈ N̂ ∩ nϕ̂. Since the
elements λ(I) form a σ-strong-∗/norm core for Λ̂, we can find a net (ωj)j∈J in I such that
λ(ωj) → x in the σ-strong-∗ topology and ξ(ωj) → Λ̂(x) in norm. Consider the net (ω̃j)j∈J .
We find λ(ω̃j) = Pγλ(ωj)Pγ → PγxPγ = x. And ξ(ω̃j) = PβPγξ(ωj) is norm convergent to
PβPγΛ̂(x). Since Λ̂ is σ-strong-∗/norm closed, it follows that Λ̂(x) = PβPγΛ̂(x) ∈ PβPγH.
Moreover, it follows from Proposition 4.2 that the range of Λ̂|N̂∩nϕ̂ is dense in PβPγH. �
We refer to [21, Section 1.1] for the definition of a GNS-representation for a C∗-algebraic
weight.
Proposition 4.4. ϕ̂\c is a proper (i.e. densely defined, lower semi-continuous, non-zero) weight
on N̂c. Its GNS-representation is given by (PγPβH, Λ̂|N̂c∩nϕ̂c , π̂|N̂c).
14 M. Caspers
Proof. By Proposition 4.2, {(ω̃⊗ ι)(W ) | ω ∈ IN} ⊆ N̂ is a norm dense subset of N̂c contained
in nϕ̂. The lower semi-continuity and non-triviality follow since ϕ̂\c is a restriction of the faithful
weight ϕ̂c. The claim on the GNS-representation follows exactly as in the proof of Proposi-
tion 4.3. �
The following lemma can be found as [18, Lemma 4.2]
Lemma 4.5. I ∩M ]
∗ is dense in M ]
∗ with respect to the norm ‖ · ‖∗.
Proposition 4.6. ϕ̂\u is a proper (i.e. densely defined, lower semi-continuous, non-zero) weight
on N̂u. Its GNS-representation is given by (PγPβH, Λ̂uιu, π̂uιu).
Proof. Lemma 4.5 shows that I ∩M ]
∗ is dense in M ]
∗. Since IN 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 IN ∩N ]
∗ is dense in N ]
∗. Hence, λ\u(IN ∩N ]
∗) is dense
in N̂u. Moreover, for ω ∈ IN ∩N ]
∗,
ϕ̂\u(λ\u(ω)∗λ\u(ω)) = ϕ̂cϑ̂ιu(λ\u(ω)∗λ\u(ω))
= ϕ̂cϑ̂(λu(ω̃)∗λu(ω̃)) = ϕ̂((ω̃ ⊗ ι)(W )∗(ω̃ ⊗ ι)(W )) <∞.
So λ\u(IN ∩ N ]
∗) is contained in n
ϕ̂\u
. Thus, ϕ̂\u is densely defined. That ϕ̂\u is lower semi-
continuous follows from [21, Definition 1.5]. Take any ω ∈ N ]
∗ such that ω̃ 6= 0, which exists
since all functionals in N∗ are given by restrictions of functionals in M∗, see Proposition 3.12.
Then, ϕ̂\u(λ\u(ω∗ ∗\ ω)) = ϕ̂u(λu(ω̃∗ ∗ ω̃)) = ϕ̂c(λ(ω̃∗ ∗ ω̃)) 6= 0, where ϕ̂c is the faithful left
invariant weight on the reduced C∗-algebraic dual (M̂c, ∆̂c).
Finally, we have to prove that Λ̂uιu maps N̂u densely into PβPγH. The proof is similar to the
one of Proposition 4.3, but since the difference is subtle we state it here. Observe that λ(I∩M ]
∗)
is a σ-strong-∗/norm core for Λ̂ as follows from [22, Proposition 2.6]. So for every x ∈ n
ϕ̂\u
we
have (ϑ̂ιu)(x) ∈ nϕ and hence there exists a net (ωj)j∈J in I ∩M ]
∗ such that λ(ωj) → (ϑ̂ιu)(x)
in the σ-strong-∗ topology and ξ(ωj) → Λ((ϑ̂ιu)(x)) = Λ̂u(ιu(x)). Consider (ω̃j)j∈J . Then,
λ(ω̃j) → (ϑ̂ιu)(x) and ξ(ω̃j) = PβPγξ(ωj) → Λ̂u(ιu(x)). Hence Λ̂u(ιu(x)) ∈ PβPγH. The range
of Λ̂uιu is dense in PβPγH by Proposition 4.2. �
Next, we like to prove that ϕ̂\ is essentially the W∗-lift of ϕ̂\u, see [21, Definition 1.31]. A priori
this question is ill-defined, since these weights are defined on different von Neumann algebras.
Indeed, ϕ̂\ is a weight on N̂ , which by definition acts on PγH, whereas the W∗-lift of ϕ̂\u is
a weight on (π̂uιu(N̂u))′′. Since Pβ ∈ M̂ ′, we see that PγPβH is an invariant subspace of N̂ . By
Proposition 4.6, the von Neumann algebra (π̂uιu(N̂u))′′ equals the restriction of N̂ to PβPγH,
i.e. (π̂uιu(N̂u))′′ = N̂Pβ acting on PβPγH.
The point is that that N̂ and N̂Pβ are in fact isomorphic. This follows in fact from Propo-
sition 4.3, but we give a different argument here. We claim, more precisely, that the map
N̂ → N̂Pβ : x 7→ xPβ is an isomorphism. Indeed, for any x in the center of M̂ , ĴxĴ = x.
Since Pβ = ĴPγ Ĵ , every projection in the center of M̂ majorizes Pβ if and only if it ma-
jorizes Pγ . Therefore, the central supports of Pβ and Pγ are equal. It follows from [11, Theo-
rem 10.3.3] that N̂ is isomorphic to PβN̂Pβ = N̂Pβ, where the isomorphism is given by the map
N̂ → N̂Pβ : x 7→ xPβ.
We emphasize that N̂ will always be considered as a von Neumann algebra acting on PγH,
whereas if we encounter (π̂uιu(N̂u))′′ we assume that it acts on PβPγH. The described iso-
morphism N̂ ' (π̂uιu(N̂u))′′, makes the following proposition well-defined. A similar argument
holds on the reduced level.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 15
Proposition 4.7. The W∗-lifts of ϕ̂\c and ϕ̂\u to N̂ both equal ϕ̂\.
Proof. We prove the proposition for ϕ̂\u, the proof for ϕ̂\c is similar. Recall from Proposition 4.3
that (PγPβH, Λ̂|N̂∩nϕ̂ , π̂|N̂ ) gives the GNS-representation of ϕ̂\. Denote the W∗-lift of ϕ̂\u by φ.
We denote its GNS-representation by (Hφ, πφ,Λφ). Recall from Proposition 4.6 that the GNS-
representation of ϕ̂\u was given by (PγPβH, Λ̂uιu, π̂uιu).
From [21, Proposition 1.32], the elements π̂uιu(x), x ∈ n
ϕ̂\u
, form a σ-strong-∗/norm core
for Λφ and we may take Hφ = PγPβH. If we can prove that the elements π̂uιu(x), x ∈ n
ϕ̂\u
,
also form a σ-strong-∗/norm core for Λ̂|N̂∩nϕ̂ , then φ and ϕ̂\ have identical GNS-representations
and hence they are equal. So let x ∈ nϕ̂ ∩ N̂ . By [22, Proposition 2.6], λ(I ∩M ]
∗) forms a σ-
strongly-∗/norm core for Λ̂. So let (ωj)j∈J be a net in I ∩M ]
∗ such that λ(ωj) → x in the
σ-strong-∗ topology and ξ(ωj) → Λ̂(x) in norm. Then, ξ(ω̃j) = PγPβξ(ωj) converges in norm.
Furthermore,
λ(ω̃j) = (ω̃j ⊗ ι)(W ) = Pγ(ωj ⊗ ι)(W )Pγ = Pγλ(ωj)Pγ → PγxPγ = x,
where the convergence is in the σ-strong-∗ topology. Note that λ(ω̃j) = (π̂uιuλ
\
u)(ωj |N ). All in
all, we conclude that (yj)j∈J := ((λ\u)(ωj |N ))j∈J is a net in n
ϕ̂\u
such that π̂uιu(yj) is σ-strong-∗
convergent to x and Λ̂|N̂∩nϕ̂(π̂uιu(yj)) is convergent. �
Remark 4.8. In particular, Proposition 4.3 implies that ϕ̂\c and ϕ̂\u are approximately KMS-
weights, see [21, Proposition 1.35].
Remark 4.9. Note that on one hand, we have a map N∗ → N̂ : ω 7→ λ(ω̃) = (ω̃ ⊗ ι)(W ). On
the other hand, we can define a map N̂∗ → N : ω 7→ λ̂(PγωPγ) = (ι⊗ ω)(1⊗ Pγ)(W ∗)(1⊗ Pγ).
These can be considered as the spherical L1-Fourier transforms. Since both ϕ\ and ϕ̂\ are
normal, semi-finite, faithful weights, one can proceed as in [1] to obtain a spherical Lp-Fourier
transform which then is a restriction of the Lp-Fourier transform as defined in [1, Theorem 5.6].
5 Spherical corepresentations
We introduce the necessary terminology for corepresentations that admit vectors that are in-
variant under the action of a quantum subgroup. These corepresentations can be considered as
spherical corepresentations.
Definition 5.1. Let U ∈ M ⊗ B(HU ) be a corepresentation. Then v ∈ HU is called a M1-
invariant vector if
(ω ⊗ ι)((Tβ ⊗ ι)(U))v = (ω ⊗ ι)(U)v, ∀ω ∈M∗.
We denote
HM1
U = {v ∈ HU | v is M1-invariant} .
Note that HM1
U is a closed subspace of HU . We denote IC(M,M1) for the equivalence classes of
irreducible corepresentations of M that admit non-trivial M1-invariant vectors. We refer to such
corepresentations as spherical corepresentations. If {(ω ⊗ ι)(U)HM1
U | ω ∈ M∗} is dense in HU ,
then we call U homogeneously cyclic. It should be clear that every irreducible corepresentation
of M that admits a non-trivial M1-invariant vector is homogeneously cyclic.
Proposition 5.2. v ∈ PγH if and only if v is M1-invariant for W .
16 M. Caspers
Proof. Observe that (Tβ⊗ ι)(W ) = W (ι⊗ π̂((ϕ1 ⊗ ι)(W1))) = W (ι⊗Pγ). This yields the only
if part. The other implication follows by taking a net (ωj)j∈J in M∗ such that λ(ωj)→ 1 in the
σ-strong-∗ topology. Then,
v ← (ωj ⊗ ι)(W )v = (ωj ⊗ ι)(Tβ ⊗ ι)(W )v = (ωj ⊗ ι)(W )Pγv → Pγv,
where the convergence is in norm. �
Lemma 5.3. Let U ∈M ⊗B(HU ) be a corepresentation. Let v ∈ HU and ω ∈ N∗. The vector
(ω̃ ⊗ ι)(U)v is M1-invariant.
Proof. This follows from the following series of equalities for which we use Lemma 3.7 (3) and
T 2
γ = Tγ . For θ ∈M∗,
(θ ⊗ ι)((Tβ ⊗ ι)(U))(ω̃ ⊗ ι)(U)v = (θTβ ⊗ ωTβTγ ⊗ ι)U13U23v
= (θTβ ⊗ ωTβTγ ⊗ ι)(∆⊗ ι)(U)v = (θ ⊗ ωTβTγ ⊗ ι)(∆⊗ ι)(U)v
= (θ ⊗ ι)(U)(ω̃ ⊗ ι)(U)v. �
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] or [5, Lemma 6.2.3]. For Proposition 5.5, one proves that
the representation N ]
∗ → B(HM1
U ) : ω → (ω̃ ⊗ ι)(U)|HM1
U
is irreducible. This can be done along
the lines of [8, Théorème III.1] or [5, Proposition 6.31].
Proposition 5.4. Let U ∈ M ⊗ B(HU ) be a corepresentation. Suppose that U admits a M1-
invariant, cyclic vector v. If dim(HM1
U ) = 1, then U is irreducible.
Proposition 5.5. Assume that N̂ is Abelian. Let U ∈ M ⊗ B(HU ) be an irreducible corepre-
sentation. Then, dim(HM1
U ) ≤ 1.
In particular, suppose that N̂ is Abelian, and that U ∈M ⊗B(HU ) is an irreducible corep-
resentation with M1-invariant unit vector v. Then x = (ι⊗ ωv,v)(U) satisfies
∆\(x) = x⊗ x. (5.1)
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 that N̂ is Abelian in order to include the example of SUq(1, 1)ext in Section 8.
Remark 5.6. Let U ∈ M ⊗ B(HU ) be a corepresentation. It follows in particular that HM1
U
is a closed invariant subspace for the representation N ]
∗ → B(HU ) : ω 7→ (ω̃ ⊗ ι)(U). By the
universal property of N̂u, we see that this gives rise to a representation of N̂u on HM1
U . Of
course, this representation can be trivial. If U is homogeneously cyclic, then the corresponding
representation of N̂u is non-degenerate. Indeed, suppose that there exist w ∈ HM1
U , such that
for all v ∈ HM1
U and ω ∈M ]
∗, 〈(ω̃ ⊗ ι)(U)v, w〉 = 0. Then, using (4) of Lemma 3.7,
0 = 〈(ωTγTβ ⊗ ι)(U)v, w〉 = 〈(ωTγ ⊗ ι)(U)v, w〉 = 〈v, (ωTγ ⊗ ι)(U)∗w〉
= 〈v, (ωTγ ⊗ ι)(U∗)w〉 = 〈v, (ωTγ ⊗ ι)(S ⊗ ι)(U)w〉
= 〈v, (ω ⊗ ι)(S ⊗ ι)(Tβ ⊗ ι)(U)w〉 = 〈v, (ω∗ ⊗ ι)(Tβ ⊗ ι)(U)w〉
= 〈v, (ω∗ ⊗ ι)(U)w〉 = 〈(ω ⊗ ι)(U)v, w〉.
We see that for all v ∈ HM1
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 of N̂u.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 17
Definition 5.7. Let U ∈ M ⊗ B(HU ) be a homogeneously cyclic corepresentation of M on
a Hilbert space HU . Then, we get a representation π\U of N̂u determined by
π\U : λ\u(ω) 7→ (ω̃ ⊗ ι)(U)|HM1
U
, ω ∈ N ]
∗.
We emphasize that the representation Hilbert space of π\U is HM1
U .
Remark 5.8. Let U ∈ M ⊗ B(HU ) 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 πU for the representation of M̂u given by λu(ω) 7→ (ω ⊗ ι)(U). Then,
π\U equals the restriction of πU ιu to HM1
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 Ux of M on a Hilbert space Hx such
that (π\Ux)x∈X is a µ-measurable field of representations of N̂u. Suppose that M̂u is separable.
Then, (Hx)x∈X is a µ-measurable field of Hilbert spaces such that (HM1
x )x∈X is a µ-measurable
field of subspaces and (Ux)x∈X is a µ-measurable field of corepresentations.
Proof. Let ωi ∈ N ]
∗, i ∈ N be a such that λu(ωi), i ∈ N is dense in M̂u. Since (π\Ux)x∈X is
µ-measurable, we have a fundamental sequence (ejx)x∈X , j ∈ N for the µ-measurable field of
Hilbert spaces (HM1
x )x∈X . For i, j ∈ N, x ∈ X, define f i,jx ∈ Hx by
f i,jx = (ωi ⊗ ι)(Ux)ejx.
We claim that (f i,jx )x∈X is a fundamental sequence for (Hx)x∈X . Indeed, since Ux is homoge-
neously cyclic, the span of (ωi ⊗ ι)(Ux)ejx, i, j ∈ N is dense in Hx. Moreover,
〈f i,jx , f i
′,j′
x 〉 = 〈(ωi ⊗ ι)(Ux)ejx, (ωi′ ⊗ ι)(Ux)ej
′
x 〉 = 〈((ω∗i′ ∗ ωi)⊗ ι)(Ux)ejx, e
j′
x 〉
= 〈((ω∗i′ ∗ ωi)∼ ⊗ ι)(Ux)ejx, e
j′
x 〉 = 〈π\Ux(λ\u((ω∗i′ ∗ ωi)|N ))ejx, e
j′
x 〉, (5.2)
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,jx )x∈X is a fundamental sequence. Moreover, by a similar computation as (5.2),
for any ω ∈M ]
∗,
〈(ω ⊗ ι)(Ux)f i,jx , f i
′,j′
x 〉 = 〈((ω∗i′ ∗ ω ∗ ωi)⊗ ι)(Ux)ejx, e
j′
x 〉
= 〈π\Ux(λ\u((ω∗i′ ∗ ω ∗ ωi)|N ))ejx, e
j′
x 〉,
is a µ-measurable function of x. For ω = ωv,w with v, w ∈ Dom(∇̂
1
2 ) ∩ Dom(∇̂−
1
2 ), we have
ω ∈ M ]
∗ by [4, Proposition 1.10]. Using [6, Proposition II.1.10 and II.2.1] it is straightforward
to prove that (Ux)x∈X is a µ-measurable field of operators. �
Proposition 5.10. Let U1 and U2 be homogeneously cyclic corepresentations of M on Hilbert
spaces H1 and H2 respectively. Suppose that the representations of N ]
∗ given by πi : ω 7→
(ω̃ ⊗ ι)(Ui)|HM1
Ui
, ω ∈ N ]
∗, i ∈ {1, 2} are equivalent. Then U1 and U2 are equivalent.
18 M. Caspers
Proof. Let T : HM1
U1
→ HM1
U2
be the unitary intertwiner between π1 and π2. Let Q0 be the
mapping{
(ω ⊗ ι)(U1)v | ω ∈M ]
∗, v ∈ H
M1
U1
}
→
{
(ω ⊗ ι)(U2)w | ω ∈M ]
∗, w ∈ H
M1
U2
}
,
(ω ⊗ ι)(U1)v 7→ (ω ⊗ ι)(U2)Tv.
This map is well-defined and isometric. Indeed, for ω ∈M ]
∗ and v ∈ HM1
U1
,
‖(ω ⊗ ι)(U2)Tv‖2 = 〈(ω∗ ∗ ω ⊗ ι)(U2)Tv, Tv〉 = 〈((ω∗ ∗ ω)∼ ⊗ ι)(U2)Tv, Tv〉
= 〈((ω∗ ∗ ω)∼ ⊗ ι)(U1)v, v〉 = 〈(ω∗ ∗ ω ⊗ ι)(U1)v, v〉 = ‖(ω ⊗ ι)(U1)v‖2,
where the second equality follows from a similar calculation as in Remark 5.6. Since U1 and U2
are homogeneously cyclic, Q0 is densely defined and has dense range. Let Q : HU1 → HU2 be
the unitary extension of Q0. Let ω, ω1, ω2 ∈M ]
∗ and v, w ∈ HM1
U1
,
〈(ω ⊗ ι)(U1)(ω1 ⊗ ι)(U1)v, (ω2 ⊗ ι)(U1)w〉 = 〈(ω∗2 ∗ ω ∗ ω1 ⊗ ι)(U1)v, w〉
= 〈((ω∗2 ∗ ω ∗ ω1)∼ ⊗ ι)(U1)v, w〉 = 〈((ω∗2 ∗ ω ∗ ω1)∼ ⊗ ι)(U2)Tv, Tw〉
= 〈(ω∗2 ∗ ω ∗ ω1 ⊗ ι)(U2)Tv, Tw〉 = 〈(ω ⊗ ι)(U2)(ω1 ⊗ ι)(U2)Tv, (ω2 ⊗ ι)(U2)Tw〉
= 〈(ω ⊗ ι)(U2)Q(ω1 ⊗ ι)(U1)v,Q(ω2 ⊗ ι)(U1)w〉,
where the second equality follows again from a similar calculation as in Remark 5.6. Since U1
is homogeneously cyclic, this proves that Q intertwines U1 with U2. �
Note that the converse of the previous proposition is clear: if U1 and U2 are equivalent corep-
resentations, then the corresponding representations as considered in Remark 5.6 are equivalent.
Remark 5.11. Proposition 5.10 and its converse also hold on the universal level. So let U1
and U2 be homogeneously cyclic corepresentations of M . π\U1
and π\U2
are equivalent if and only
if U1 and U2 are equivalent.
6 Representations of M̂u, M̂c, N̂u and N̂c
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 of N̂c ‘lift’ to representations of M̂c.
Let us give a more elaborate discussion. There are three special types of representations
within Rep(N̂u).
1. As explained in Remark 5.6, the corepresentations of M give rise to representations
of N̂u. Recall [18] that the corepresentations of M are in 1-to-1 correspondence with
non-degenerate representations of M̂u. Hence, the representations of M̂u give rise to rep-
resentations of N̂u. This correspondence can be described more directly: if π is a repre-
sentation of M̂u on a Hilbert space Hπ, then πιu is the corresponding representation of N̂u
on the closure of ((πιu)(N̂u))Hπ. Note that by Remark 5.11, this assignment descends to
a well-defined, injective map on the equivalence classes of representations,
Rep(M̂u) ↪→ Rep(N̂u) : π 7→ πιu.
2. If π is a representation of N̂c, then πϑ̂\ is a representation of N̂u. These representations
correspond to the representations that are weakly contained in the GNS-representation
of ϕ̂\u. Indeed this follows, since by Proposition 4.6, this GNS-representation is given by
π̂uιu = π̂cϑ̂ιu = π̂cϑ̂
\ = π̂|N̂c ϑ̂
\.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 19
Hence, every representation πϑ̂\, with π ∈ Rep(N̂c) is weakly contained in the GNS-
representation of N̂u. The other way around, any representation of N̂u that is weakly
contained in the GNS-representation of ϕ̂\u factors through the canonical projection ϑ̂\.
3. If π is a representation of M̂c, then πϑ̂\ = πϑ̂ιu is a representation of N̂u. Here, we used
that N̂c ⊆ M̂c.
The main results of this section will be the following. We prove that every representation
of N̂c comes from a representation of M̂c, i.e. the representations of N̂u obtained in (2) and (3)
are the same ones.
Theorem 6.1. For every non-degenerate representation ρ ∈ Rep(N̂c), there exists a non-
degenerate representation π ∈ Rep(M̂c) on a Hilbert space Hπ such that ρ is equivalent to
the restriction of π|N̂c to the closure of π(N̂c)Hπ.
Proof. Let M ],β
∗ denote space of functionals θ ∈M ]
∗ such that θTβ = θ. By a similar argument
as in the proof of Proposition 3.13, we see that if θ ∈M ]
∗, then θTβ ∈M ],β
∗ and (θTβ)∗ = θ∗Tγ .
In particular, (N ]
∗)
∼ ⊆M ],β
∗ . Note that for θ1, θ2 ∈M ],β
∗ , we have that (θ∗1 ∗ θ2)∼ = θ∗1 ∗ θ2.
We complete M ],β
∗ into a (right) Hilbert N̂c-module. For θ, θ1, θ2 ∈M ],β
∗ and ω ∈ N ]
∗, we put
θ · ω = θ ∗ ω̃ ∈M ],β
∗ , (6.1)
〈θ1, θ2〉N]
∗
= (θ∗1 ∗ θ2)|N ∈ N ]
∗. (6.2)
The fact that (6.1) is in M ],β
∗ follows from Lemma 3.7. That (6.2) is in N ]
∗ follows from Propo-
sition 3.13. This gives a right N ]
∗-module structure on M ],β
∗ . We will apply [27, Lemma 2.16]
to get a Hilbert N̂c-module X. Here we consider N ]
∗ is a subalgebra of N̂c by means of the
map λ\. To continue, note that conditions (a) and (c) of [27, Definition 2.1] are indeed satisfied.
(b) follows, since for θ1, θ2 ∈M ],β
∗ and ω ∈ N ]
∗,
〈θ1, θ2 · ω〉N]
∗
= (θ∗1 ⊗ θ2 ⊗ ω)(ι⊗ ι⊗ TβTγ)(∆⊗ ι)∆|N ,
〈θ1, θ2〉N]
∗
∗\ ω = (θ∗1 ⊗ θ2)∆|N ∗\ ω = (θ∗1 ⊗ θ2 ⊗ ω)(ι⊗ ι⊗ Tγ)(∆⊗ ι)∆|N .
Since (ι⊗ Tβ)∆|N = ∆Tβ|N = ∆|N , these expressions are equal. (d) follows, since for θ ∈M ],β
∗ ,
λ\((θ∗ ∗ θ)|N ) = λ(θ∗ ∗ θ) = λ(θ)∗λ(θ) ≥ 0,
in N̂c. This defines the right Hilbert N̂c-module X and we denote its norm by ‖ · ‖X .
We are able to define an action of M ]
∗ on X by means of adjointable operators which extends
the convolution product on M ]
∗. Indeed, for ω ∈M ]
∗, and θ1, θ2 ∈M ],β
∗ ,
〈ω ∗ θ1, θ2〉N]
∗
= θ∗1 ∗ ω∗ ∗ θ2|N = 〈θ1, ω
∗ ∗ θ2〉N]
∗
.
Furthermore, for ω ∈M ]
∗ and θ ∈M ],β
∗ ,
‖ω ∗ θ‖2X = ‖λ(θ∗ ∗ ω∗ ∗ ω ∗ θ)‖ ≤ ‖λ(ω)‖2‖λ(θ∗ ∗ θ)‖ = ‖λ(ω)‖2‖θ‖2X . (6.3)
So the action of ω ∈M ]
∗ is bounded. This allows us to extend the action of ω to an adjointable
operator on X. We denote this operator by Lω. Moreover, from (6.3) we get a representation
of M̂c on the Hilbert module X by means of adjointable operators. This representation is
uniquely determined by λ\(ω) 7→ Lω, ω ∈ N ]
∗.
Now, let ρ be a representation of N̂c on a Hilbert space Hρ. By [27, Proposition 2.66] we get
an induced representation of M̂c on X⊗N̂cHρ. Let us denote the latter by Ind ρ. Let (aj)j∈J be
20 M. Caspers
an approximate unit for M̂c. Then, for θ ∈M ],β
∗ , v ∈ Hρ, we see that (Ind ρ)(aj)(θ⊗v)→ (θ⊗v)
in X. So Ind ρ is non-degenerate.
Note that the completion of (N ]
∗)
∼⊗N̂cHρ is a closed subspace of the Hilbert space X⊗N̂cHρ
that is isomorphic to Hρ via the unitary extension T of
(N ]
∗)
∼ ⊗N̂c Hρ → Hρ : ω̃ ⊗ v 7→ ρ(λ\(ω))v. (6.4)
The map (6.4) extends unitarily since ρ is non-degenerate. For ω, θ ∈ N ]
∗, v ∈ Hρ,
(Ind ρ)(λ\(ω))(θ̃ ⊗ v) = ω̃ ∗ θ̃ ⊗ v = (ω ∗\ θ)∼ ⊗ v,
so that (N ]
∗)
∼ ⊗Hρ is an invariant subspace for Ind ρ. We denote its closure by Y . Moreover,
for ω, θ ∈ N ]
∗ and v ∈ Hρ,
T (Ind ρ)(λ\(ω))(θ̃ ⊗ v) = T ((ω̃ ∗ θ̃)⊗ v) = T ((ω ∗\ θ)∼ ⊗ v)
= ρ(λ\(ω ∗\ θ))v = ρ(λ\(ω))ρ(λ\(θ))v = ρ(λ\(ω))T (θ̃ ⊗ v),
so that T intertwines (Ind ρ)|N̂c restricted to the Hilbert space Y with ρ. Finally, we claim
that Y equals the closure of (Ind ρ)(N̂c)X. Indeed, for any ω ∈ N ]
∗, θ ∈ M ],β
∗ and v ∈ Hρ, we
see that (Ind ρ)(λ\(ω))(θ⊗ v) = (ω̃ ∗ θ)⊗ v = (ω ∗\ θ|N )∼⊗ v ∈ Y . Since N̂cN̂c ⊆ N̂c is dense, it
is straightforward to prove that Y equals the closure of (Ind ρ)(N̂c)X in X. This concludes the
proof by choosing π = Ind ρ. �
Corollary 6.2. For every representation ρ of N̂u that factors through ϑ̂\, there is a homoge-
neously 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 of N̂u corresponds. Recall
that the GNS-representation of ϕ̂\u was given by the triple (PγPβH, Λ̂uιu, π̂uιu), see Proposi-
tion 4.6. Since π̂uιu = π̂|N̂c ϑ̂
\, we can apply Corollary 6.2. We define the closed subspace
E = {(ω̃ ⊗ ι)(W )PβPγH | ω ∈ N ]
∗} = {(π̂uιuλ\u(ω)PβPγH | ω ∈ N ]
∗} ⊆ H, (6.5)
where the closure is with respect to the norm in H. It is clear that the representation of N̂u
that corresponds to the restriction of W to E equals π̂uιu.
We use the notation IR(M̂u,M1) to denote the irreducible representations π of M̂u such that
the representation πιu is non-trivial. Under the 1-1 correspondence between IR(M̂u) and IC(M),
see [18], we see from Remark 5.11 and the remarks following Definition 5.7 that IR(M̂u,M1)
corresponds to IC(M,M1). Let IR(M̂c,M1) denote the irreducible representations of M̂c such
that the restriction to N̂c is non-trivial. We find the following diagram of inclusions.
IR(N̂u) ←↩ IR(N̂c)
IC(M,M1) ' IR(M̂u,M1)
?�
OO
←↩ IR(M̂c,M1)
?�
'
OO
(6.6)
Note that the map IR(M̂u,M1) ↪→ IR(N̂u) indeed maps into the representations of N̂u 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(M̂c,M1) ↪→ IR(M̂u,M1) is not surjec-
tive. We briefly comment on the fact that also the inclusion IR(M̂u,M1) ↪→ IR(N̂u) is generally
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 21
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-K-
invariant functions forms a representation, i.e. a homomorphism that preserves the ∗-operation,
whereas the representation of all L1-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 ιu : N̂u → M̂u 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(N̂u), there is a representation π ∈ Rep(M̂u) on a Hilbert space Hπ,
such that ρ is equivalent to the restriction of πιu to (πιu(N̂u))Hπ.
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 ιu plays an essential role at two
places. First of all, the injectivity of ιu can be used to prove positivity of the inner product (6.2),
since in this case an element in N̂u is positive if and only if it is positive in M̂u. Secondly, the
universal analogue of (6.3) can be recovered from the injectivity of ιu, since for ω ∈ M ]
∗ and
θ ∈M ],β
∗ ,
‖λ\u(θ∗ ∗ ω∗ ∗ ω ∗ θ)‖\u = ‖λu(θ∗ ∗ ω∗ ∗ ω ∗ θ)‖u
≤ ‖λu(ω∗ ∗ ω)‖u‖λu(θ∗ ∗ θ)‖u = ‖λu(ω)‖u‖λ\u(θ∗ ∗ θ)‖\u.
The rest of the prove of Theorem 6.1 can be copied mutatis mutandis.
7 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∗-algebra N̂u 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 M1-invariant vector.
Recall that we defined the space E in (6.5). Let L be any Hilbert space and let L0 ⊆ L
be a closed subspace. We denote the conjugate Hilbert space of L by L. Note that the space
L⊗L0 can canonically be identified with the Hilbert-Schmidt operators in B(L0,L). We denote
the latter space by B2(L0,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.
Theorem 7.1 (Plancherel–Godement). Let (M,∆) be a unimodular locally compact quantum
group and let (M1,∆1) be a compact (closed) quantum subgroup. Let N̂ and N̂u be the von
Neumann algebra and C∗-algebra as defined earlier in this section. Suppose that N̂ is a type I
von Neumann algebra and that N̂u and M̂u are separable.
Then, there exists a standard measure µM1 on IC(M,M1), a µM1-measurable field of Hilbert
spaces (HU )U∈IC(M,M1) of which (HM1
U )U∈IC(M,M1) forms a measurable field of subspaces, a mea-
surable field of self-adjoint, strictly positive operators (DM1
U )U∈IC(M,M1) acting on HM1
U and an
isomorphism QM1 of E onto
∫ ⊕
IC(M,M1)HU ⊗H
M1
U dµM1(U) with properties:
1. For ω ∈ IN and µM1-almost all U ∈ IC(M,M1), the operator (ω̃ ⊗ ι)(U)(DM1
U )−1 is
bounded and (ω̃ ⊗ ι)(U) · (DM1
U )−1 is in B2(HM1
U ).
22 M. Caspers
2. For ω1, ω2 ∈ IN , we have
〈ξ(ω̃1), ξ(ω̃2)〉
=
∫
IC(M,M1)
Tr
(
((ω̃2 ⊗ ι)(U) · (DM1
U )−1)∗((ω̃1 ⊗ ι)(U) · (DM1
U )−1)
)
dµM1(U),
and we let QM1
0 : PβPγH →
∫ ⊕
IC(M,M1)H
M1
U ⊗HM1
U dµM1(U) be the isometric extension of
Λ̂(λ(ĨN ))→
∫ ⊕
IC(M,M1)
B2(HM1
U )dµ(U) :
ξ(ω̃) 7→
∫ ⊕
IC(M,M1)
(ω̃ ⊗ ι)(U) · (DM1
U )−1dµM1(U).
3. QM1
0 intertwines π̂uιu and
∫ ⊕
IC(M,M1) π
\
U ⊗ 1
HM1
U
dµM1(U).
4. QM1 intertwines the restriction of W to E with
∫ ⊕
IC(M,M1) U ⊗ 1
HM1
U
dµM1(U). Moreover,
the restriction of QM1 to PβPγH →
∫ ⊕
IC(M,M1)H
M1
U ⊗HM1
U dµM1(U) equals QM1
0 .
5. Assume moreover that M̂ is a type I von Neumann algebra and that M̂u is separable. Let
µ, DU , Q be defined as in [4, Theorem 3.4.1]. Then HM1
U is an invariant subspace for
µ-almost all DU and IC(M,M1) is a µ-measurable subset of IC(M). If one takes:
• µM1 equal to the restriction of µ to IC(M,M1);
• DM1
U equal to the restriction of DU to HM1
U ;
• QM1 the restriction of Q to E.
Then, µM1, DM1
U , QM1 satisfy the properties (1)–(4).
Proof. By Propositions 4.6 and 4.7, ϕ̂\u is a proper approximate KMS-weight. Therefore, we
can apply [4, Theorem 3.4.5], so that we obtain a measure µM1 on IR(N̂u), a measurable field of
Hilbert spaces (KM1
σ )σ∈IR(N̂u), a measurable field of representations (πσ)σ∈IR(N̂u), a measurable
field of self-adjoint, strictly positive operators (DM1
σ )σ∈IR(N̂u) and an isomorphismQM1
0 of PγPβH
onto
∫ ⊕
IR(N̂u)
KM1
σ ⊗KM1
σ dµM1(σ) satisfying the properties of this theorem.
Let ρ ∈ IR(N̂u) be in the support of µM1 . We claim that πρ is weakly contained in π̂uιu.
Suppose that this is not the case, so that there exists x ∈ N̂u such that πρ(x) 6= 0 but π̂uιu(x) = 0.
Let X = {σ ∈ IR(N̂u) | πσ(x) 6= 0}. Then X is an open neighbourhood of ρ. Moreover, it follows
form [4, Theorem 3.4.5] that QM1
0 intertwines π̂uιu with
∫ ⊕
IR(N̂u)
πσ ⊗ 1Kσdµ
M1(σ) from which
it follows that µM1(X) = 0. This contradicts the fact that ρ is in the support of µM1 , so πρ is
weakly contained in π̂uιu.
Since π̂uιu = π̂|N̂c ϑ̂
\, we see that ρ is in IR(N̂c), where IR(N̂c) is considered as a subset
of IR(N̂u) by the inclusion (6.6). Now we use Theorem 6.1 to identify IR(N̂c) with IR(M̂c,M1),
which we consider as a subspace of IC(M,M1) by (6.6). We consider µM1 as a measure
on IC(M,M1) by defining the complement of IR(N̂c) in IC(M,M1) to be negligible. Let
Uσ ∈ IC(M,M1) denote the corepresentation corresponding to σ ∈ IR(N̂c). So, πσ = π\Uσ .
We write DM1
Uσ
for DM1
σ and set DU = 0 for U ∈ IC(M,M1) not in the support of µM1 . We
denote HU for the corepresentation Hilbert space of U ∈ IC(M,M1), and we get HM1
Uσ
= KM1
σ .
Therefore, since the support of µ is contained in IR(N̂c), we see that QM1
0 is a map from
PβPγH →
∫ ⊕
IC(M,M1)H
M1
U ⊗HM1
U dµM1(U).
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 23
(1). It follows from Corollary 6.2, Remark 5.6 and the properties of DM1
σ described in (1) of
[4, Theorem 3.4.5], that for ω ∈ IN , the operator (ω̃⊗ ι)(U)(DM1
U )−1 is bounded and its closure
is Hilbert-Schmidt for µM1-almost every U ∈ IC(M,M1).
(2) and (3). We make two observations. First note that by Proposition 4.6, for ω ∈ IN ,
Λ̂uιu(λ\u(ω)) = ξ(ω̃). Secondly, we have proved that for every ρ in the support of µM1 , there
is a U ∈ IC(M,M1), such that πρ = 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 (HU )U∈IC(M,M1) is a measurable field of Hilbert spaces of which (HM1
U )U∈IC(M,M1) forms
a measurable field of subspaces. Here we used that M̂u is separable.
To prove (4), we make the following observations. First of all, by Remark 6.3, we see that
π̂uιu = π\WE , where WE denotes the restriction of the multiplicative unitary W to the Hilbert
space E . Secondly,∫ ⊕
IC(M,M1)
π\U ⊗ 1
HM1
U
dµM1(U) = π\∫⊕
IC(M,M1)
U⊗1
HM1
U
dµ(U)
,
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 WE and
∫ ⊕
IC(M,M1) U ⊗ 1
HM1
U
dµM1(U) are equivalent. Moreover,
we define QM1 to be the intertwiner as constructed in the proof of Proposition 5.10. Then,
QM1 satisfies (4).
We now prove (5). The proof of Proposition 3.16 shows that σ̂t(Pγ) = Pγ . Write D =∫ ⊕
IC(M)DUdµ(U). Since D−2 is the Radon–Nikodym derivative of ϕ̂ with respect to a trace
on M̂ , see the proof of [4, Theorem 3.4.5], we have DηM̂ and σ̂t(x) = D−2itxD2it. Thus, Pγ
commutes with Dit for all t ∈ R. Since Pγ ∈ M̂ '
∫ ⊕
IC(M)B(HU )dµ(U), we have a direct
integral decomposition Pγ =
∫ ⊕
IC(M)(Pγ)Udµ(U) and (Pγ)U commutes with Dit
U for µ-almost all
U ∈ IC(M).
Pγ is the projection of H onto HM1 , see Proposition 5.2. A vector v =
∫ ⊕
IC(M) vUdµ(U) is
M1-invariant for W if and only if vU is M1-invariant for µ-almost all U ∈ IC(M), as follows
directly from Definition 5.1. Hence (Pγ)U is the projection of HU onto HM1
U for µ-almost all
U ∈ IC(M). We record three conclusions:
(P1) HM1
U is an invariant subspace of DU for µ-almost all U ∈ IC(M);
(P2) IC(M,M1) is a µ-measurable subset of IC(M) by [6, Proposition II.1.1 (i)];
(P3) The image of PγH under Q equals
∫ ⊕
IC(M,M1)H
M1
U ⊗HUdµ(U).
For the choice of µM1 , DM1
U and QM1 made in (5), properties (1) and (2) follow from
the properties of µ, DU and Q as described in [4, Theorem 3.4.1] using Proposition 4.2.
By [4, Theorem 3.4.1], Q intertwines W with
∫ ⊕
IC(M) U ⊗ 1HUdµ(U). Using (P3) together
with [4, Theorem 3.4.5 (3)] and the fact that Pβ = ĴPγ Ĵ , we see that Q restricts to a uni-
tary map from PβPγH to
∫ ⊕
IC(M,M1)H
M1
U ⊗ HM1
U dµ(U). Hence Q restricts to a unitary map
from E to
∫ ⊕
IC(M,M1)HU ⊗ H
M1
U dµ(U), which then intertwines the restriction of W to E with∫ ⊕
IC(M,M1) U ⊗ 1
HM1
U
dµ(U). This proves (4), from which (3) follows by the construction in Re-
marks 5.6 and 6.3. �
Remark 7.2. If ∆\ is cocommutative, then N̂ is Abelian. By Proposition 5.5 we see that for
all U ∈ IC(M), dim(HM1
U ) ≤ 1. Hence the operators DM1
U are scalars. We may assume that
DM1
U = 1 by replacing the measure µM1 if necessary. In particular, we see that for a classical
Gelfand pair (G,K), the map QM1
0 is the spherical Fourier transform.
24 M. Caspers
Remark 7.3. The support of µM1 is given by IR(N̂c). Here, IR(N̂c) is a subspace of IC(M,M1)
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 µM1 is contained in IR(N̂c).
Remark 7.4. As pointed out in remark [2, Remark 3.3], the Theorem 7.1 also holds if one
assumes that N̂c and M̂c are separable, instead of N̂u. Moreover, note that if M̂c is separable,
then so is N̂c ⊆ M̂c. If M̂ is a type I von Neumann algebra, then so is N̂ = PγM̂Pγ . In
particular, if M̂ is type I and M̂c is separable, then then the result of Theorem 7.1 holds for any
closed quantum subgroup of (M,∆).
8 Example: SUq(1, 1)ext
Let (M,∆) be the quantum group analogue of the normaliser of SU(1, 1) in SL(2,C) as intro-
duced 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 and N̂ as defined in the previous section can be equipped with a Z2-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 Z2-grading. Moreover, we
derive product formulae for little q-Jacobi functions appearing as matrix coefficients of corep-
resentations which admit invariant vectors.
Remark 8.1. In [2, Propositions B.2 and B.3] it is proved that SUq(1, 1)ext satisfies the hy-
potheses of the Plancherel theorem, [4, Theorem 3.4.1]. By Remark 7.4 we can also apply
Theorem 7.1.
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 δx 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 Iq = −qN ∪ qZ, 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
W =
⊕
p∈qZ
∫ ⊕
[−1,1]
Wp,xdx⊕
⊕
x∈σd(Ωp)
Wp,x
. (8.1)
Here σd(Ωp) 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]. Wp,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 =
∫ ⊕
Wp,xd(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 Wp,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.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 25
Using the notation of [10, Sections 10.2 and 10.3], an orthonormal basis for the corepre-
sentation Hilbert space Lp,x of the principal and complementary series Wp,x is given by the
vectors
eε,ηm (p, x), ε, η ∈ {−,+}, m ∈ Z. (8.2)
For the discrete series corepresentation Wp,x a subset of the vectors (8.2) gives a basis for the
corepresentation space Lp,x, see [10, Proposition 5.2]. For our purposes, it is convenient to use
the following notational convention.
Notation 8.3. Let p ∈ qZ, x ∈ σd(Ωp), so that Wp,x is a discrete series corepresentation. We
denote eε,ηm (p, x), ε, η ∈ {−,+},m ∈ Z for the zero vector in case eε,ηm (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 Lp,x.
For any p ∈ qZ, x ∈ (µ(−q), µ(q)) ∪ σd(Ωp), we set f ε,ηm (p, x) = eε,η
m− 1
2
χ(p)
(p, x), ε, η ∈ {−,+},
where m ∈ Z if p ∈ q2Z and m ∈ 1
2 + Z if p ∈ q1+2Z. We remind the reader that the direct
integrals over of the vectors f ε,ηm (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
Kfε,ηm (p, x) = qmf ε,ηm (p, x),
(q−1 − q)Efε,ηm (p, x) = q−
1
2
−m|1 + εηq2m+1eiθ|f ε,ηm+1(p, x),
U+−
0 f ε,ηm (p, x) = η(−1)υ(p)f ε,−ηm (p, x),
U−+
0 f ε,ηm (p, x) = εηχ(p)(−1)m−
1
2
χ(p)f−ε,ηm (p, x), (8.3)
where θ is such that x = µ(eiθ). Similar expressions can be obtained for the discrete and com-
plementary 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 ∈ qZ, x ∈ µ(−q2Z+1p∪q2Z+1p), one can define a corepresentation Wp,x
by defining the action of the generators of M̂ by means of [10, Lemma 10.1]. Since the actions
of the generators of Wpr,x are equal for any r ∈ q4Z, 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
W '
⊕
p∈qZ
∫ ⊕
[−1,1]
Wp,xdx⊕
⊕
x∈µ(−q2Z+1p∪q2Z+1p)
Wp,x
.
We will also need the following expressions for the matrix coefficients. By [10, Lemma 10.9
and Section 10.3], for p ∈ qZ, x ∈ (µ(−q), µ(q)),
(ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
) (Wp,x) fm0,p0,t0 (8.4)
= C
(
εηx;m′ − 1
2χ(p), ε′, η′; εε′|p0|q−m−m
′
, p0,m−m′
)
δsgn(p0),ηη′fm0−m+m′,εε′|p0|q−m−m′ ,t0 .
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.
26 M. Caspers
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 opera-
tor K equals 0 ∪ q
1
2
Z.
Definition 8.6. We define a normal, injective ∗-homomorphism π̂ : L∞(Z) → M̂ by setting
π̂(δk) = δ
q
k
2
(K).
Note that π̂ preserves the comultiplication, since
(π̂ ⊗ π̂)∆Z(δk) = (π̂ ⊗ π̂)
(∑
l∈Z
δl ⊗ δk−l
)
=
∑
l∈Z
δ
q
l
2
(K)⊗ δ
q
k−l
2
(K) = δ
q
k
2
(K ⊗K) = δ
q
k
2
(∆̂(K)).
Therefore, π̂ identifies (T,∆T) as a closed quantum subgroup of (M,∆). Furthermore, π̂ induces
a morphism π between the universal quantum groups (Mu,∆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.
Proposition 8.7. For all p ∈ qZ, x ∈ (µ(−q), µ(q)) ∪ σd(Ωp),
β
(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x) =
(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x)⊗ ζ2m, (8.5)
γ
(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x) =
(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x)⊗ ζ2m′ . (8.6)
Here, ζ is the identity function on the complex unit circle T.
Proof. Note that π̂ has a direct integral decomposition π̂ =
∫ ⊕
π̂p,xd(p, x). Here π̂p,x :
L∞(Z) → B(Lp,x) is determined by π̂p,x(δk) = δ
q
k
2
(K), where the action of K on the rep-
resentation 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∫ ⊕
(β ⊗ ι)(Wp,x)d(p, x) =
∫ ⊕
(Wp,x)13(ι⊗ π̂p,x)(WT)23d(p, x).
By (8.3) and the definition of π̂, for almost all pairs (p, x) in the decomposition (8.1),
(β ⊗ ι)(Wp,x) = (Wp,x)13(ι⊗ π̂p,x)(WT)23. (8.7)
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 ∈ qZ,
(µ(−q), µ(q))→M ⊗ L∞(T) : x 7→ β
((
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x)
)
, (8.8)
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 27
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 ∈ qZ,
(µ(−q), µ(q))→M ⊗ L∞(T) : x 7→
(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x)⊗ ζ2m, (8.9)
extends to an analytic function on a neighbourhood of (µ(−q), µ(q)). Note that for all p ∈ qZ,
x ∈ (µ(−q), µ(q)),(
ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
(Wp,x)⊗ ζ2m
=
(
ι⊗ ι⊗ ω
fε,ηm (p,x),fε
′,η′
m′ (p,x)
)
((Wp,x)13(ι⊗ π̂p,x)(WT)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 map π̂ on the von Neumann algebraic level since L∞(T) is not separable. Therefore, we
have defined the representations π̂p,x explicitly.
Recall that we defined L∞(T)-invariant vectors in Definition 5.1.
Corollary 8.9. Let p ∈ qZ, x ∈ (µ(−q), µ(q)) ∪ σd(Ωp).
1. If p ∈ q2Z, then the space of L∞(T)-invariant vectors of Wp,x is spanned by f ε,η0 (p, x), with
ε, η ∈ {+,−}.
2. If p ∈ q2Z, x ∈ σd(Ωp). The space of L∞(T)-invariant vectors of Wp,x is one dimensional.
3. If p ∈ q1+2Z, then Wp,x has no L∞(T)-invariant vectors.
4. Pγ = δ1(K) = π̂(δ0) and hence the von Neumann algebras N and N̂ that are constructed
in Section 3 are given by
N =
{
(ι⊗ ωδ1(K)v,δ1(K)w)(W ) | v, w ∈ K
}σ-strong-∗
, N̂ = δ1(K)M̂δ1(K).
Proof. From the considerations in Sections 3 and 5, (1), (3) and (4) follow. For (2) consider
λ ∈ −q2Z+1 ∪ q2Z+1 be such that x = µ(λ) and |λ| ≥ 1. Put j′, l′ ∈ Z by setting |λ| = q1−2j′ =
q1+2l′ . In particular, l′ < j′. For case (i) of [10, Proposition 5.2], f−,+0 (p, x) is zero if and only if
0 > l′ if and only if |λ| ≥ 1, which is true by assumption. Similarly, f+,−
0 (p, x) = 0. Hence the
only L∞(T)-invariant vector is f+,+
0 (p, x). Cases (ii) and (iii) of [10, Proposition 5.2] follow in
exactly the same manner. �
Remark 8.10. Note that the space of L∞(T)-invariant vectors for the irreducible components
of Wp,x is not necessarily of dimension ≤ 1. For example Wp,x, p ∈ q2Z, x ∈ [−1, 1]\{0} splits
as a sum of 2 irreducible corepresentations of which the L∞(T)-invariant vectors form a 2-
dimensional vector space, see [10, Section 10.2]. This implies that ∆\ is not cocommutative and
we can not use (5.1) directly to obtain product formulae. However, we will define gradings on
the spaces N and N̂ which still allows us to find such formulae.
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
28 M. Caspers
are important to determine the spaces N , N̂ and the spherical Fourier transform, see Theo-
rem 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 ∈ q2Z,
x ∈ (µ(−q), µ(q)), set
g1,+
m (p, x) =
1
2
√
2(f+,+
m (p, x) + iχ(p)f−,−m (p, x)),
g1,−
m (p, x) =
1
2
√
2(f+,−
m (p, x)− iχ(p)f−,+m (p, x)),
g2,+
m (p, x) =
1
2
√
2(f+,+
m (p, x)− iχ(p)f−,−m (p, x)),
g2,−
m (p, x) =
1
2
√
2(f+,−
m (p, x) + iχ(p)f−,+m (p, x)).
For every j ∈ {1, 2}, p ∈ q2Z, x ∈ (µ(−q), µ(q)), the vectors gj,σm (p, x), σ ∈ {+,−}, m ∈ Z form
an orthonormal basis for Ljp,x, the corepresentation space of one of the summands of Wp,x, see
[10, equation (95)]. For p ∈ q2Z, x ∈ σp(Ωd), set
g1,+
m (p, x) = f+,+
m (p, x) + iχ(p)f−,−m (p, x),
g1,−
m (p, x) = f+,−
m (p, x)− iχ(p)f−,+m (p, x),
g2,+
m (p, x) = f+,+
m (p, x)− iχ(p)f−,−m (p, x),
g2,−
m (p, x) = f+,−
m (p, x) + iχ(p)f−,+m (p, x). (8.10)
Recall that we made the convention that f ε,ηm (p, x) = 0 in case f ε,ηm (p, x) is not in the basis given
in [10, Proposition 5.2]. Hence, for x ∈ σd(Ωp), 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,ηm (p, x) for
some ε, η ∈ {+,−} modulo a phase factor.
Now, we determine which of the discrete, principal and complementary series corepresenta-
tions 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.
Proposition 8.11.
1. Let x, x′ ∈ (µ(−q), µ(q))\{0}, p, p′ ∈ q2Z, j, j′ ∈ {1, 2}. W j
p,x ' W j′
p′,x′ if and only if either
x = ±x′, j = j′, p/p′ ∈ q4Z or x = ±x′, j 6= j′, p/p′ ∈ q2+4Z.
2. Let p, p′ ∈ q2Z, j, j′, k, k′ ∈ {1, 2}. W j,k
p,0 ' W j′,k′
p′,0 if and only if either j = j′, k = k′,
p/p′ ∈ q4Z or j 6= j′, k = k′, p/p′ ∈ q2+4Z.
3. Let x, x′ ∈ µ(−qZ ∪ qZ), p, p′ ∈ q2Z. Wp,x 'Wp′,x′ if and only if |x| = |x′|.
Proof. The proposition follows from a careful comparison of the action of the generators,
see (8.3) for the principal series, [10, Proposition 5.2 and Lemma 10.1] for the discrete series
and [10, Section 10.3] for the complementary series. We prove (1). By considering the Casimir
operator [10, Definition 4.5] one sees that if an irreducible component of Wp,x is equivalent to
an irreducible component of Wp′,x′ , then |x| = |x′|.
In case x = x′, an intertwiner must send gj,±m (p, x) to a non-zero scalar multiple of gj
′,±
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:
(i) p/p′ ∈ q4Z, j = j′, for which it sends gj,±m (p, x) to gj,±m (p′, x);
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 29
(ii) p/p′ ∈ q2+4Z, j 6= j′, in which case it sends gj,±m (p, x) to ±gj
′,±
m (p′, x).
Similarly, in case x = −x′, an intertwiner must send gj,±m (p, x) to gj
′,∓
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′ ∈ q4Z, j = j′ for which it sends gj,±m (p, x) to ±gj,∓m (p′,−x);
(ii) p/p′ ∈ q2+4Z, j 6= j′, in which case it sends gj,±m (p, x) to gj
′,±
m (p′,−x).
This proves (1), the other cases follows similarly. �
Summarizing Corollary 8.9 and Proposition 8.11, we find that IC(M,M1), the space of equiva-
lence classes of irreducible spherical corepresentations is partly given by
IC(M,M1)⊇ (0, 1, 1) ∪ (0, 1, 2) ∪ (0, 2, 1) ∪ (0, 2, 2) ∪ ((0, 1]× {1, 2}) (principal)
∪ µ(q2N+1) (discrete)
∪ (1, µ(q))× {1, 2} (complementary)
(8.11)
Here, we identify the points (0, 1, 1), (0, 1, 2), (0, 2, 1), (0, 2, 2) with the respective irreducible
corepresentations W 1,1
1,0 , W 1,2
1,0 , W 2,1
1,0 , W 2,2
1,0 , see [10, Proposition 10.14 (ii)]. We let a point
(x, j) ∈ (0, 1]× {1, 2} correspond to W j
1,x, see [10, Proposition 10.12]. The points x ∈ µ(q2N+1)
corresponds to W1,x, see [10, Proposition 5.2]. We emphasize that it is not known if the corep-
resentations described in [10] are all the corepresentations, therefore we do not know if this
description completely describes IC(M,M1). For the von Neumann algebras N and N̂ 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
··
··
· · · . . .
Figure 1. Known part of IC(M,M1) for SUq(1, 1)ext with the diagonal subgroup, c.f. (8.11). Part I:
Principal spherical series. Part II: Complementary spherical series. Part III: Discrete spherical series.
The von Neumann algebra N
The next step is to make N and N̂ more explicit and meanwhile define a grading on these spaces.
Therefore, we will first find an alternative formula for the mapping TβTγ 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
1
2
Z. For k, l ∈ Z, set
Mk,l =
{
(ι⊗ ωδ
q
1
2 (k−l) (K)v,δ
q
1
2 (k+l)
(K)w)(W ) | v, w ∈ K
}σ-strong-∗
.
By (8.3) and (8.4), the spaces Mk,l have mutually trivial intersection. Moreover, M equals the
σ-strong-∗ closure of the direct sum of vector spaces ⊕k,l∈ZMk,l. Note that N = M0,0.
30 M. Caspers
Definition 8.12. For m ∈ Z, p, t ∈ −qZ ∪ qZ, let Pm,p,t be the orthogonal rank one projection
of K onto the space spanned by the vector fm,p,t. Here we define fm,p,t to be the zero vector if
either p 6∈ Iq or t 6∈ Iq. Define mappings T+ and T− defined by
T± : M →M : x 7→
∑
m∈Z,p,t∈Iq
Pm,±p,txPm,p,t,
where the sum converges in the strong topology.
Proposition 8.13. T+ + T− = TβTγ.
Proof. For a normal functional ω =
∑
i∈I ωξi,ηi ∈ M∗, with
∑
i∈I ‖ξi‖2,
∑
i∈I ‖ηi‖2 < ∞, the
linear map
M → C : x 7→ ωT±(x) =
∑
i∈I
∑
m∈Z,p,t∈Iq
〈xPm,p,tξi, Pm,±p,tηi〉,
is normal. Hence, the maps T± are normal. Moreover, for x ∈Mk,l, using (8.4),
T+(x) + T−(x) =
{
x, if k = l = 0,
0, otherwise.
(8.12)
For x ∈ Mk,l we see by using (8.5) and (8.6) that TβTγ(x) also equals the right hand side
of (8.12). Since also TβTγ is normal, this proves that T+ + T− = TβTγ . �
Let u0 ∈ B(L2(Iq)) be the partial isometry determined by u0 : δp 7→ δ−p, and recall that u
was defined as u = 1⊗ u0 ∈ B(L2(Z)⊗ L2(Iq)). By [13, Lemma 2.4] M ' L∞(T)⊗B(L2(Iq)).
Proposition 8.13 yields,
N = TβTγ(M) = (T+ + T−)(M),
which is isomorphic to the von Neumann subalgebra of M generated by 1 ⊗ L∞(Iq) and u.
Therefore, introduce the following identifications.
Definition 8.14. We identifyN with the von Neumann algebra acting on L2(Iq) being generated
by L∞(Iq) and u0. We split N as a direct sum of vector spaces
N = N+ ⊕N−, where N+ = L∞(Iq), and N− = L∞(Iq)u0 = L∞(Iq ∩ (−1, 1))u0.
This turns N into a Z2-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:
ϕ\(f) =
∑
p0∈Iq
f(p0)p2
0, f ∈ N+ = L∞(Iq)
+.
For f ∈ m+
ϕ\
we find that u0f ∈ mϕ\ and it follows by [13, Definition 4.1] that ϕ\(u0f) = 0, so
that Λ(nϕ∩N+) and Λ(nϕ∩N−) are orthogonal spaces. The discussion so far allows us to make
the following identifications.
Definition 8.15. Identify the closure of Λ(nϕ∩N+) with L2(Iq). We will write L2(N+) for L2(Iq)
to indicate explicitly that L2(Iq) is considered as part of the GNS-space of ϕ\. Similarly, we
identify the closure of Λ(nϕ ∩ N−) with L2(Iq ∩ (−1, 1)), by identifying fu0 ∈ N−, where
f ∈ L∞(Iq∩(−1, 1))∩L2(Iq∩(−1, 1)) with f ∈ L2(Iq∩(−1, 1)). We write short hand L2(N−) for
L2(Iq∩(−1, 1)). We write L2(N) = L2(N+)⊕L2(N−). We emphasize that here the spaces L2(Iq)
(respectively L2(Iq ∩ (−1, 1))) should be understood with respect to the integral given by the
weighted sum
∑
p0∈Iq( · )p2
0 (respectively
∑
p0∈Iq∩(−1,1)( · )p2
0).
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 31
The von Neumann algebra N̂
We now turn our attention to the von Neumann algebra N̂ as defined in Section 3. Considered
as a subalgebra of M̂ , it inherits the Z2-grading defined in [10, Definition 4.7].
Definition 8.16. We set
N̂+ = N̂ ∩ M̂+, N̂− = N̂ ∩ M̂−,
see [10, Definition 4.7] for M̂+ and M̂−. This turns N̂ into a Z2-graded algebra.
Proposition 8.17. We have an isomorphism of von Neumann algebras:
N̂ '
∫ ⊕
[0,1]
M2(C)⊕M2(C)dx⊕
⊕
x∈µ(q2N+1)
C. (8.13)
The isomorphism is determined by the map
(ω̃ ⊗ ι)(W ) 7→
∫ ⊕
[0,1]
(ω̃ ⊗ ι)(W 1
1,x)⊕ (ω̃ ⊗ ι)(W 2
1,x)dx⊕
⊕
x∈µ(q2N+1)
(ω̃ ⊗ ι)(W1,x). (8.14)
Under this isomorphism N̂+ corresponds to the direct integrals over matrices whose entries vanish
off the diagonal. N̂− 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 ∈ M̂ ∩ M̂ ′ be the projection as in the proof of [2, Proposition B.2]. By
considering the Casimir operator, one sees that P is the spectral projection on the interval
[−1, 1] of the Casimir operator. In [2, Proposition B.2], it is shown that
PM̂P '
∫ ⊕
x∈[0,1]
M̂xdx,
where M̂x is generated by {(ω ⊗ ι)(Wx) | ω ∈M∗} and Wx = (⊕p∈qZWp,x)⊕ (⊕p∈qZWp,−x). By
similar techniques as in [2, Proposition B.2], one can show that
M̂ '
∫ ⊕
x∈[0,1]
M̂xdx⊕
⊕
x∈σd(Ω)∩(1,∞)
M̂x, (8.15)
where M̂x, x ∈ σd(Ω) ∩ (1,∞) is generated by {(ω ⊗ ι)(Wx) | ω ∈M∗} and
Wx = (
⊕
p∈qZ,s.t.x∈σd(Ωp)
Wp,x)⊕ (
⊕
p∈qZ,s.t.−x∈σd(Ωp)
Wp,−x).
Since Pγ ∈ M̂ , it has a direct integral decomposition as in (8.15), i.e.
PγM̂Pγ '
∫ ⊕
x∈[0,1]
(Pγ)xM̂x(Pγ)xdx⊕
⊕
x∈σd(Ω)∩(1,∞)
(Pγ)xM̂x(Pγ)x.
Similarly, K decomposes with respect to the direct integral decomposition (8.15) as a direct
integral over a field (Kx)x∈[0,1]∪(σd(Ω)∩(1,∞)) and by Proposition 8.7, we see that (TβTγ⊗ι)(Wx) =
(1⊗ δ1(Kx))Wx(1⊗ δ1(Kx)) = (1⊗ (Pγ)x)Wx(1⊗ (Pγ)x). Hence, (Pγ)xM̂x(Pγ)x is generated by
{(ω̃ ⊗ ι)(Wx) | ω ∈ M∗}. Due to Corollary 8.9 and Proposition 8.11, the latter von Neumann
algebra is isomorphic to M2(C)⊕M2(C) in case x ∈ (0, 1] and to C in case x ∈ σd(Ω) ∩ (1,∞).
This proves (8.13). That (8.14) gives the isomorphism follows directly from this proof. The
claim on the gradings follows from [10, Proposition 4.8]. �
32 M. Caspers
Remark 8.18. The von Neumann algebra M̂0 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 decom-
position (8.13), and to avoid some redundant extra notation we have not treated the point x = 0
separately.
We claim that ϕ̂\, i.e. the restriction of the dual left Haar weight to N̂ , is a trace. Indeed, it
follows from [2, Section 5] and [4, Proposition 3.5.5] that for any p ∈ qZ, x ∈ [−1, 1] ∪ σd(Ωp),
there is a constant c(p, x) such that
Dp,xf
ε,η
m (p, x) = p1/2qmc(p, x)f ε,ηm (p, x).
Here D =
∫ ⊕
Dp,xd(p, x), with short hand notation Dp,x = DWp,x , is the Duflo–Moore opera-
tor arising from the Plancherel theorem, see [4, Theorem 3.4.1] and also Theorem 7.1. Since
the eigenvalues of Dp,x are independent of the signs ε, η, we see that
∫ ⊕
D
L∞(T)
p,x d(p, x) =
Pγ
∫ ⊕
Dp,xd(p, x)Pγ is central in N̂ . Since
∫ ⊕
(D
L∞(T)
p,x )−2d(p, x) is the Radon–Nikodym deriva-
tive of ϕ̂\ with respect to a trace, see the proof of [4, Theorem 3.4.5], we see that ϕ̂\ is a trace.
Therefore, under the identification (8.13), we see that
ϕ̂\ =
∫ ⊕
[0,1]
(
TrM2(C) ⊕ TrM2(C)
)
d(x)dx⊕
⊕
x∈µ(q2N+1)
d(x)TrC, (8.16)
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
L2(N̂) =
∫ ⊕
[0,1]
M2(C)⊕M2(C)dx⊕
⊕
x∈µ(q2N+1)
C,
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)(TrM2(C) ⊕ TrM2(C)) for the integral part and d(x)TrC for the direct summands.
It follows from (8.13) and (8.16) that Λ̂(nϕ̂ ∩ N̂+) and Λ̂(nϕ̂ ∩ N̂−) are orthogonal spaces, we
denote their closures interpreted within L2(N̂) by L2(N̂+) and L2(N̂−). They consist of direct
integrals of diagonal matrices and off-diagonal matrices, respectively.
The spherical Fourier transform
Here, we determine the spherical Fourier transform, i.e. we determine the map QL
∞(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 GNS-
spaces for ϕ\ and ϕ̂\ with L2(N) and L2(N̂), we may consider QL
∞(T)
0 as a map from L2(N)
to L2(N̂) and use the short hand notation F2 : L2(N)→ L2(N̂) for this map.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 33
If f ∈ L1(Iq) ∩ L∞(Iq) ⊆ 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,
F2 : Λ(f) 7→ ((f · ϕ\)∼ ⊗ ι)
∫ ⊕
[0,1]
W1,xdx⊕
⊕
x∈µ(q2N+1)
W1,x
. (8.17)
Here, the right hand side indeed is an element of L2(N̂) by Corollary 8.9. Note that we choose
the inner product on L2(N̂) to be the direct integral over the traces d(x)(TrM2(C) ⊕ TrM2(C))
for the integral part and d(x)TrC 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 ∈ L1(Iq)∩L∞(Iq),
so that fu0 ∈ N−,
F2 : Λ(fu0) 7→ ((fu0 · ϕ\)∼ ⊗ ι)
∫ ⊕
[0,1]
W1,xdx⊕
⊕
x∈µ(q2N+1)
W1,x
. (8.18)
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 W1,x in its
kernel. The next step is to make these kernels explicit.
For any m0 ∈ Z, t0 ∈ Iq, x ∈ [−1, 1] ∪ µ(−q2N+1 ∪ q2N+1), j ∈ {1, 2}, σ, τ ∈ {+,−}, put
Kσ,τ
j (p0;x) = 〈(ι⊗ ω
gj,σ0 (1,x),gj,τ0 (1,x)
)(W1,x)fm0,p0,t0 , fm0,στp0,t0〉.
We emphasize that Kσ,τ
j (p0;x) for x not in σd(Ω1) is defined by Remark 8.5. This expression is
independent of m0 and t0. Due to Proposition 8.11, we have the following symmetry
Kσ,τ
j (p0;x) = στK−σ,−τj (p0;−x). (8.19)
From (8.4), (8.17) and (8.18), we see that we get a graded Fourier transform F2,+ ⊕ F2,− :
L2(N+) ⊕ L2(N−) → L2(N̂+) ⊕ L2(N̂−), which is defined by sending f ⊕ gu0 ∈ (L2(Iq) ∩
L1(Iq)) ⊕ (L2(Iq ∩ (−1, 1)) ∩ L1(Iq ∩ (−1, 1)))u0 ⊆ L2(N+) ⊕ L2(N−) to the matrix valued
function on IC(M,L∞(T)) determined by sending (x, j) ∈ [0, 1]× {0, 1} to∑
p0∈Iq
(
K+,+
j (p0;x)f(p0) K−,+j (p0;x)g(p0)
K+,−
j (p0;x)g(p0) K−,−j (p0;x)f(p0)
)
p2
0, (8.20)
and x ∈ µ(q2Z+1) to∑
p0∈Iq
K+,+
0 (p0;x)f(p0)p2
0 =
∑
p0∈Iq
K+,+
1 (p0;x)f(p0)p2
0.
Note that for x = 0, the matrix appearing in (8.20) is actually a direct sum of two matrix blocks
after a basis transformation, since W1,0 splits as a direct sum of four irreducible corepresenta-
tions, see [10, Proposition 10.13]. By Theorem 7.1, this map is unitary.
For completeness, we give the analogous result for the inverse Fourier transform F−1
2 :
L2(N̂+) ⊕ L2(N̂−) → L2(N+) ⊕ L2(N−). Since QL
∞(T)
0 is a restriction of the Plancherel trans-
formation Q, see Theorem 7.1, we see that F−1
2 can be considered as the restriction of Q−1.
The transform Q−1 is described in [1] on the operator algebraic level, see in particular [1, Ex-
ample 3.13]. See also [41] for the algebraic counterpart. More explicitly, the spherical inverse
Fourier transform is determined by
F−1
2 : f 7→ (ι⊗ (f · ϕ̂))(W ∗),
34 M. Caspers
where f ∈ N̂ ∩ nϕ̂ is such that there is a normal functional on M̂ , denoted by (f · ϕ̂), which
is determined by (f · ϕ̂)(x) = ϕ̂(xf), x ∈ n∗ϕ̂. By the decomposition of W (8.1), we find the
following theorem.
Theorem 8.20. For σ, τ ∈ {+,−}, let fσ,τ1 , fσ,τ2 ∈ L1([0, 1]) ∩ L2([0, 1]), where L1([0, 1]) and
L2([0, 1]) should be understood with respect to the integral
∫
[0,1] d(x)dx. Let fd ∈ L1(µ(q2N+1))∩
L2(µ(q2N+1)), where L1(µ(q2N+1)) and L2(µ(q2N+1)) should be understood with respect to the
integral
∑
x∈µ(q2N+1) d(x). Define the function f ∈ L2(N̂) by
f(x) =
(
f+,+
1 (x) f−,+1 (x)
f+,−
1 (x) f−,−1 (x)
)
⊕
(
f+,+
2 (x) f−,+2 (x)
f+,−
2 (x) f−,−2 (x)
)
, x ∈ [0, 1],
fd(x), x ∈ µ(q2N+1).
Then,∫
[0,1]
(
f+,+
1 (x)K+,+
1 (p0;x) + f−,−1 (x)K−,−1 (p0;x)
)
d(x)dx
+
∫
[0,1]
(
f+,+
2 (x)K+,+
2 (p0;x) + f−,−2 (x)K−,−2 (p0;x)
)
d(x)dx
+
∑
x∈µ(q2N+1)
(
fd(x)K+,+
1 (p0;x)d(x)
)
⊕(∫
[0,1]
(
f−,+1 (x)K+,−
1 (p0;x) + f+,−
1 (x)K−,+1 (p0;x)
)
d(x)dx
+
∫
[0,1]
(
f−,+2 (x)K+,−
2 (p0;x) + f+,−
2 (x)K−,+2 (p0;x)
)
d(x)dx
)
u0 (8.21)
exists for every p0 ∈ Iq. Moreover, (8.21) considered as a direct sum of functions in p0 forms
an element of L2(N+)⊕L2(N−). This mapping extends to a unitary map L2(N̂+)⊕L2(N̂−)→
L2(N+)⊕ L2(N−), which is inverse to F2.
We explicitly state the formulae for the kernels Kσ,τ
j (p0;x) for x ∈ [−1, 1]∪µ(−q2N+1∪q2N+1),
which can be expressed in terms of little q-Jacobi functions. Using the notation of [10, Section 9]
for S(·) and A(·), we find
K+,+
1 (p0;x) = S(−λ, p0, p0, 0)×
1, p0 < 0,
A(λ, 1, 0,+,+)
A(λ, 1, 0,−,−)
, p0 > 0,
K+,+
2 (p0;x) = S(−λ, p0, p0, 0)×
1, p0 < 0,
−A(λ, 1, 0,+,+)
A(λ, 1, 0,−,−)
, p0 > 0,
K+,−
1 (p0;x) = −S(λ,−p0, p0, 0)×
A(λ, 1, 0,+,+)
A(−λ, 1, 0,+,−)
, p0 < 0,
− A(λ, 1, 0,+,+)
A(−λ, 1, 0,−,+)
, p0 > 0,
K+,−
2 (p0;x) = −S(λ,−p0, p0, 0)×
A(λ, 1, 0,+,+)
A(−λ, 1, 0,+,−)
, p0 < 0,
A(λ, 1, 0,+,+)
A(−λ, 1, 0,−,+)
, p0 > 0.
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 35
The fractions of the functions A(·) are phase factors. Here, λ ∈ T such that µ(λ) = x. And, for
λ ∈ C\{0}, simplifying [10, Lemma 9.1 (ii)],
S(±λ,∓p0, p0, 0) = |p0|2ν(p0)2c2
q
√
(±κ(p0),−κ(p0); q2)∞(∓q2; q2)∞
× (q2,−q2/κ(p0), λq2, 1/λ,−q/λ; q2)∞
(sgn(p0)1/λ, sgn(p0)λq2,±q/λ; q2)∞
2ϕ1
(
−q/λ,−λq
−q2
; q2,−q2/κ(p0)
)
.
The other kernels occurring in (8.20) can be expressed explicitly by means of the symmetry
relation (8.19). The matrix coefficients Kσ,τ
j (p0, x) are special types of little q-Jacobi functions,
see also [10, Appendix B.5] and references given there.
Product formulae for little q-Jacobi functions
Let N∗,+ be the space of normal functionals in N∗ which are zero on N−. Let N∗,− be the space
of normal functionals in N∗ which are zero on N+. Note that ϕ\ is a trace on N . Therefore, every
functional in N∗,+ is given by f · ϕ\, where f is a function on Iq so that
∑
p0∈Iq f(p0)p2
0 < ∞.
Every functional in N∗,− is given by fu0 · ϕ\, where f is a function on Iq ∩ (−1, 1) so that∑
p0∈Iq∩(−1,1) f(p0)p2
0 <∞. So N∗ = N∗,+ ⊕N∗,− as vector spaces.
For p ∈ Iq, recall that δp denotes the function on Iq whose value is 1 in p and 0 elsewhere.
We write δp,+ for δp · ϕ\ ∈ N∗,+. For p ∈ Iq, we write δp,− for the functional δpu0 · ϕ\ ∈ N∗,−.
Note that only for p ∈ Iq ∩ (−1, 1), this functional is non-zero, but it is convenient to keep this
notation.
Remark 8.21. Note that if one identifies N with the subalgebra (1 ⊗ L∞(Iq) ⊗ 1) ∪ (1 ⊗
L∞(Iq)u0 ⊗ 1) of M acting on the GNS-space K, then δp,+ = ωfm0,p,t0 ,fm0,p,t0
and δp,− =
ωfm0,−p,t0 ,fm0,p,t0
, where m0 ∈ Z, t0 ∈ Iq. This functional is independent of m0 and t0 if conside-
red as a functional on N .
Using the fact that the Fourier transform preserves the Z2-gradings on N and N̂ , we see that
for p1, p2 ∈ Iq:
(δp1,+ ⊗ δp2,+)∆\ ∈ N∗,+, (δp1,− ⊗ δp2,−)∆\ ∈ N∗,+,
(δp1,+ ⊗ δp2,−)∆\ ∈ N∗,−, (δp1,− ⊗ δp2,+)∆\ ∈ N∗,−.
Hence, we see that for f, g ∈ L∞(Iq), there exist constants Ap0(p1, p2), Bp0(p1, p2), p0, p1, p2 ∈ Iq
and Cp0(p1, p2), Dp0(p1, p2), p0 ∈ Iq ∩ (−1, 1), p1, p2 ∈ Iq such that for any p1, p2 ∈ Iq the four
sums ∑
p0∈Iq
|Ap0(p1, p2)|p2
0,
∑
p0∈Iq
|Bp0(p1, p2)|p2
0,∑
p0∈Iq∩(−1,1)
|Cp0(p1, p2)|p2
0,
∑
p0∈Iq∩(−1,1)
|Dp0(p1, p2)|p2
0,
are finite and
(δp1,+ ⊗ δp2,+)∆\(f) =
∑
p0∈Iq
Ap0(p1, p2)f(p0)p2
0, (8.22)
(δp1,− ⊗ δp2,−)∆\(f) =
∑
p0∈Iq
Bp0(p1, p2)f(p0)p2
0, (8.23)
(δp1,+ ⊗ δp2,−)∆\(fu0) =
∑
p0∈Iq∩(−1,1)
Cp0(p1, p2)f(p0)p2
0, (8.24)
36 M. Caspers
(δp1,− ⊗ δp2,+)∆\(fu0) =
∑
p0∈Iq∩(−1,1)
Dp0(p1, p2)f(p0)p2
0. (8.25)
Putting f = Kσ,τ
j (p0;x) for any x ∈ [−1, 0) ∪ (0, 1] ∪ µ(−q2N+1 ∪ q2N+1), j ∈ {1, 2}, this yields
a product formula for 2ϕ1-series. For p1, p2 ∈ Iq and p3, p4 ∈ Iq ∩ (−1, 1),
K+,+
j (p1;x)K+,+
j (p2;x) =
∑
p0∈Iq
Ap0(p1, p2)K+,+
j (p0;x)p2
0, (8.26)
K+,−
j (p3;x)K−,+j (p4;x) =
∑
p0∈Iq
Bp0(p3, p4)K−,−j (p0;x)p2
0, (8.27)
K+,+
j (p1;x)K−,+j (p3;x) =
∑
p0∈Iq∩(−1,1)
Cp0(p1, p3)K−,+j (p0;x)p2
0, (8.28)
K−,+j (p3;x)K−,−j (p1;x) =
∑
p0∈Iq∩(−1,1)
Dp0(p3, p1)K−,+j (p0;x)p2
0. (8.29)
Remark 8.22. Note that the gradings on N and N̂ make that the left hand sides of (8.26)–(8.29)
consists of a single product of two 2ϕ1-functions.
Remark 8.23. If x ∈ µ(−q2N+1∪q2N+1), so that W1,x is a discrete series corepresentation, then
equations (8.27)–(8.29) are trivial, i.e. they equate 0 to 0. If in addition x < 0, then (8.26) is
also trivial.
Remark 8.24. From (8.22)–(8.25), we can also get formulae for the products
K−,−j (p1;x)K−,−j (p2;x), K+,−
j (p3;x)K−,+j (p4;x),
K+,+
j (p1;x)K−,+j (p3;x), K−,+j (p3;x)K−,−j (p1;x),
where j ∈ {1, 2}, p1, p2 ∈ Iq and p3, p4 ∈ Iq∩(−1, 1). However, using the symmetry (8.19), these
formulae are already contained in (8.26)–(8.29).
In the remainder of this section we determine the coefficient functions A, B, C, and D. We
explicitly show how to find A, the other coefficients can be found by the same method. Let
f = δp0 , so that the right hand side of (8.22) is equal to Ap0(p1, p2)p2
0. To determine the left
hand side, note that δp0 = δsgn(p0)(e)δp−2
0
(γ∗γ). Recall that ∆(e) = e ⊗ e ∈ N ⊗ N , so that
∆(δsgn(p0)(e)) = δsgn(p0)(e⊗ e) ∈ N ⊗N . Hence,
∆\(δsgn(p0)(e)) = ∆(δsgn(p0)(e)) = δ1(e)⊗ δsgn(p0)(e) + δ−1(e)⊗ δ−sgn(p0)(e) ∈ N ⊗N.
Note that by the relation ∆(x) = W ∗(1 ⊗ x)W , x ∈ M , we find that ∆(δp−2
0
(γ∗γ)) equals the
projection onto the closure of
span
{
W ∗fm′,p′,t′ ⊗ fm,p0,t | m,m′ ∈ Z, p′, t, t′ ∈ Iq
}
.
This projection is given by the formula
K ⊗K → K⊗K : v 7→
∑
m,m′∈Z,p′,t,t′∈Iq
〈v,W ∗fm′,p′,t′ ⊗ fm,p0,t〉W ∗fm′,p′,t′ ⊗ fm,p0,t,
where the sum is norm convergent. Note that for x ∈ N , ∆(x) ∈ Mγ ⊗Mβ, so that ∆\(x) =
(ι ⊗ TβTγ)∆(x) = (ι ⊗ (T+ + T−))∆(x). Since ∆\(δp−2
0
(γ∗γ)) ∈ N ⊗ N , we find that for any
m1,m2 ∈ Z and t1, t2 ∈ Iq,
(δp1 ⊗ δp2)∆\(δp−2
0
(γ∗γ))
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 37
= 〈(ι⊗ (T+ + T−))(∆(δp−2
0
(γ∗γ)))fm1,p1,t1 ⊗ fm2,p2,t2 , fm1,p1,t1 ⊗ fm2,p2,t2〉
= 〈∆(δp−2
0
(γ∗γ))fm1,p1,t1 ⊗ fm2,p2,t2 , fm1,p1,t1 ⊗ fm2,p2,t2〉
=
∑
m′,m∈Z,p′,t,t′∈Iq
|〈fm1,p1,t1 ⊗ fm2,p2,t2 ,W
∗fm′,p′,t′ ⊗ fm,p0,t〉|2, (8.30)
where the equations follow from Remark 8.21 and Proposition 8.13, the definition of T+ and T−
and the discussion above. Recall the functions ap(x, y), p, x, y ∈ Iq from [10, Definition 6.2].
From [13, Propositions 4.5 and 4.10], we see that (8.30) equals
∑
t∈Iq
(
t
t2
)2
at(sgn(p0p2t2)p1q
m2t, t2)2ap0(p1, p2)2
=
∑
t∈Iq
at2(sgn(t2)|p1|qm2t, t)2ap0(p1, p2)2 = ap0(p1, p2)2.
Here, the first equality follows from [10, equation (24)] and the fact that by definition ap0(p1, p2)2
= 0 if sgn(p0p2) 6= sgn(p1). The last equality follows from [10, Proposition 6.3]. Hence, using
[30, Section IX.4, equation (4)] in the second equality,
Ap0(p1, p2)p2
0 = (δp1 ⊗ δp2)∆\
(
δsgn(p0)(e)δp−2
0
(γ∗γ)
)
= (δp1 ⊗ δp2)(ι⊗ Tγ)
(
δsgn(p0)(∆(e))δp−2
0
∆(γ∗γ)
)
= (δp1 ⊗ δp2)δsgn(p0)(∆(e))(ι⊗ Tγ)
(
δp−2
0
∆(γ∗γ)
)
= δsgn(p0),sgn(p1p2)ap0(p1, p2)2 = ap0(p1, p2)2.
Recall [13, Definition 3.1] that by definition ap0(p1, p2) = 0 if δsgn(p0),sgn(p1p2) = 0, so indeed the
last equality follows. By a similar computation we get,
Bp0(p1, p2)p2
0 = ap0(p1, p2)ap0(−p1,−p2),
Cp0(p1, p2)p2
0 = ap0(p1,−p2)a−p0(p1, p2),
Dp0(p1, p2)p2
0 = ap0(−p1, p2)a−p0(p1, p2).
Remark 8.25. In particular, the sums in (8.26)–(8.29) run only through either the positive or
the negative numbers, since az(x, y) = 0 if sgn(xyz) = −1.
Remark 8.26. It is also possible to obtain (8.22)–(8.25) from [10, Proposition 4.10], using the
pairing between M and {λ(ω) | ω ∈M∗} ⊆ M̂ , defined by 〈x, λ(ω)〉 = ω(x).
Remark 8.27. In case of the group SU(1, 1) we know [43, Chapter 6] that there exists an
addition formula corresponding to the product formula, i.e. the product formula corresponds to
the constant term in the addition formula. It would be of interest to obtain addition formulae
corresponding to (8.26)–(8.29).
Acknowledgement
The author likes to thank Erik Koelink for the useful discussions and Noud Aldenhoven for
providing Fig. 1. Also, the author benefits from a detailed referee report.
38 M. Caspers
References
[1] Caspers M., The Lp-Fourier transform on locally compact quantum groups, J. Operator Theory, to appear,
arXiv:1008.2603.
[2] Caspers M., Koelink E., Modular properties of matrix coefficients of corepresentations of a locally compact
quantum group, J. Lie Theory 21 (2011), 905–928, arXiv:1003.2278.
[3] De Commer K., On a correspondence between quantum SU(2), quantum E(2) and extended quantum
SU(1, 1), Comm. Math. Phys. 304 (2011), 187–228, arXiv:1004.4307.
[4] Desmedt P., Aspects of the theory of locally compact quantum groups: Amenability–Plancherel measure,
Ph.D. Thesis, Katholieke Universiteit Leuven, 2003.
[5] van Dijk G., Introduction to harmonic analysis and generalized Gelfand pairs, de Gruyter Studies in Math-
ematics, Vol. 36, Walter de Gruyter & Co., Berlin, 2009.
[6] Dixmier J., Les algèbres d’opérateurs dans l’espace hilbertien, Gauthiers-Villars, Paris, 1957.
[7] Dixmier J., Les C∗-algèbres et leurs représentations, Gauthiers-Villars, Paris, 1969.
[8] Faraut J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, in Analyse Har-
monique (Universite de Nancy I, 1980), Editors J.L. Clerc et al., Les Cours du C.I.M.P.A., Centre Interna-
tional de Mathematiques Pures et Appliquees, Nice, France, 1983, 315–446.
[9] Floris P.G.A., Gel’fand pair criteria for compact matrix quantum groups, Indag. Math. (N.S.) 6 (1995),
83–98.
[10] Groenevelt W., Koelink E., Kustermans J., The dual quantum group for the quantum group analogue of the
normalizer of SU(1, 1) in SL(2,C), Int. Math. Res. Not. 2010 (2010), no. 7, 1167–1314, arXiv:0905.2830.
[11] Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras, Vol. II, Advanced theory,
Graduate Studies in Mathematics, Vol. 16, American Mathematical Society, Providence, RI, 1997.
[12] Kasprzak P., On a certain approach to quantum homogeneous spaces, arXiv:1007.2438.
[13] Koelink E., Kustermans J., A locally compact quantum group analogue of the normalizer of SU(1, 1) in
SL(2,C), Comm. Math. Phys. 233 (2003), 231–296, math.QA/0105117.
[14] Koelink E., Stokman J.V., Fourier transforms on the quantum SU(1, 1) group, With an appendix by Mizan
Rahman, Publ. Res. Inst. Math. Sci. 37 (2001), 621–715, math.QA/9911163.
[15] Koelink H.T., Askey–Wilson polynomials and the quantum SU(2) group: survey and applications, Acta
Appl. Math. 44 (1996), 295–352.
[16] Koelink H.T., The quantum group of plane motions and the Hahn–Exton q-Bessel function, Duke Math. J.
76 (1994), 483–508.
[17] Koornwinder T.H., Positive convolution structures associated with quantum groups, in Probability Measures
on Groups, X (Oberwolfach, 1990), Plenum, New York, 1991, 249–268.
[18] Kustermans J., Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001),
289–338, math.OA/9902015.
[19] Kustermans J., Locally compact quantum groups, in Quantum Independent Increment Processes. I, Lecture
Notes in Math., Vol. 1865, Springer, Berlin, 2005, 99–180.
[20] Kustermans J., One-parameter representations of C∗-algebras, funct-an/9707009.
[21] Kustermans J., Vaes S., Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000),
837–934.
[22] Kustermans J., Vaes S., Locally compact quantum groups in the von Neumann algebraic setting, Math.
Scand. 92 (2003), 68–92, math.OA/0005219.
[23] Lance C., Direct integrals of left Hilbert algebras, Math. Ann. 216 (1975), 11–28.
[24] Noumi M., Yamada H., Mimachi K., Finite-dimensional representations of the quantum group GLq(n,C)
and the zonal spherical functions on Uq(n− 1)\Uq(n), Japan. J. Math. (N.S.) 19 (1993), 31–80.
[25] Nussbaum A.E., Reduction theory for unbounded closed operators in Hilbert space, Duke Math. J. 31
(1964), 33–44.
[26] Podkolzin G.B., Vainerman L.I., Quantum Stiefel manifold and double cosets of quantum unitary group,
Pacific J. Math. 188 (1999), 179–199.
[27] Raeburn I., Williams D.P., Morita equivalence and continuous-trace C∗-algebras, Mathematical Surveys and
Monographs, Vol. 60, American Mathematical Society, Providence, RI, 1998.
http://arxiv.org/abs/1008.2603
http://arxiv.org/abs/1003.2278
http://dx.doi.org/10.1007/s00220-011-1208-y
http://arxiv.org/abs/1004.4307
http://dx.doi.org/10.1515/9783110220209
http://dx.doi.org/10.1515/9783110220209
http://dx.doi.org/10.1016/0019-3577(95)98202-M
http://dx.doi.org/10.1093/imrn/rnp173
http://arxiv.org/abs/0905.2830
http://arxiv.org/abs/1007.2438
http://dx.doi.org/10.1007/s00220-002-0736-x
http://arxiv.org/abs/math.QA/0105117
http://dx.doi.org/10.2977/prims/1145477332
http://arxiv.org/abs/math.QA/9911163
http://dx.doi.org/10.1007/BF00047396
http://dx.doi.org/10.1007/BF00047396
http://dx.doi.org/10.1215/S0012-7094-94-07617-5
http://dx.doi.org/10.1142/S0129167X01000757
http://arxiv.org/abs/math.OA/9902015
http://dx.doi.org/10.1007/11376569_2
http://dx.doi.org/10.1007/11376569_2
http://arxiv.org/abs/funct-an/9707009
http://dx.doi.org/10.1016/S0012-9593(00)01055-7
http://arxiv.org/abs/math.OA/0005219
http://dx.doi.org/10.1007/BF02547968
http://dx.doi.org/10.1215/S0012-7094-64-03103-5
Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs 39
[28] Salmi P., Skalski A., Idempotent states on locally compact quantum groups, Quart. J. Math., to appear,
arXiv:1102.2051.
[29] Takesaki M., Theory of operator algebras. I, Springer-Verlag, New York – Heidelberg, 1979.
[30] Takesaki M., Theory of operator algebras. II, Springer-Verlag, Berlin, 2003.
[31] Timmermann T., An invitation to quantum groups and duality. From Hopf algebras to multiplicative uni-
taries and beyond, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008.
[32] Tomatsu R., A characterization of right coideals of quotient type and its application to classification of
Poisson boundaries, Comm. Math. Phys. 275 (2007), 271–296, math.OA/0611327.
[33] Vaes S., A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2005), 317–374,
math.OA/0407525.
[34] Vaes S., A Radon–Nikodym theorem for von Neumann algebras, J. Operator Theory 46 (2001), 477–489,
math.OA/9811122.
[35] Vaes S., The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180 (2001),
426–480, math.OA/0005262.
[36] Vaes S., Vainerman L.I., Extensions of locally compact quantum groups and the bicrossed product construc-
tion, Adv. Math. 175 (2003), 1–101, math.OA/0101133.
[37] Vaes S., Vainerman L.I., On low-dimensional locally compact quantum groups, in Locally Compact Quantum
Groups and Groupoids (Strasbourg, 2002), IRMA Lect. Math. Theor. Phys., Vol. 2, de Gruyter, Berlin, 2003,
127–187, math.QA/0207271.
[38] Vainerman L.I., Gel’fand pair associated with the quantum group of motions of the plane and q-Bessel
functions, Rep. Math. Phys. 35 (1995), 303–326.
[39] Vainerman L.I., Gel’fand pairs of quantum groups, hypergroups and q-special functions, in Applications of
Hypergroups and Related Measure Algebras (Seattle, WA, 1993), Contemp. Math., Vol. 183, Amer. Math.
Soc., Providence, RI, 1995, 373–394.
[40] Vainerman L.I., Hypergroup structures associated with Gel’fand pairs of compact quantum groups, in Recent
Advances in Operator Algebras (Orléans, 1992), Asterisque (1995), no. 232, 231–242.
[41] Van Daele A., The Fourier transform in quantum group theory, in New Techniques in Hopf Algebras
and Graded Ring Theory, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2007, 187–196,
math.RA/0609502.
[42] Van Daele A., Locally compact quantum groups. A von Neumann algebra approach, math.OA/0602212.
[43] Vilenkin N.Ja., Klimyk A.U., Representations of Lie groups and special functions, Vol. 1, Simplest Lie
groups, special functions and integral transforms, Kluwer Academic Publishers Group, 1991.
[44] Vogan D.A., Representations of real reductive Lie groups, Progress in Mathematics, Vol. 15, Birkhäuser,
Boston, Mass., 1981.
http://dx.doi.org/10.1093/qmath/har023
http://arxiv.org/abs/1102.2051
http://dx.doi.org/10.4171/043
http://dx.doi.org/10.1007/s00220-007-0267-6
http://arxiv.org/abs/math.OA/0611327
http://dx.doi.org/10.1016/j.jfa.2004.11.016
http://arxiv.org/abs/math.OA/0407525
http://arxiv.org/abs/math.OA/9811122
http://dx.doi.org/10.1006/jfan.2000.3704
http://arxiv.org/abs/math.OA/0005262
http://dx.doi.org/10.1016/S0001-8708(02)00040-3
http://arxiv.org/abs/math.OA/0101133
http://arxiv.org/abs/math.QA/0207271
http://dx.doi.org/10.1016/0034-4877(96)89291-3
http://arxiv.org/abs/math.RA/0609502
http://arxiv.org/abs/math.OA/06022121
1 Introduction
2 Preliminaries and notation
3 Spherical Fourier transforms
4 Weights on homogeneous spaces
5 Spherical corepresentations
6 Representations of u, c, u and c
7 A quantum group analogue of the Plancherel-Godement theorem
8 Example: SUq(1,1)ext
References
|
| id | nasplib_isofts_kiev_ua-123456789-147385 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:15:48Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Caspers, M. 2019-02-14T16:55:15Z 2019-02-14T16:55:15Z 2011 Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs / M. Caspers // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16T99; 43A90 https://nasplib.isofts.kiev.ua/handle/123456789/147385 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,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 Z₂-grading, we obtain product formulae for little q-Jacobi functions. This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html. The author likes to thank Erik Koelink for the useful discussions and Noud Aldenhoven for providing Fig. 1. Also, the author benefits from a detailed referee report. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs Article published earlier |
| spellingShingle | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs Caspers, M. |
| title | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs |
| title_full | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs |
| title_fullStr | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs |
| title_full_unstemmed | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs |
| title_short | Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs |
| title_sort | spherical fourier transforms on locally compact quantum gelfand pairs |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147385 |
| work_keys_str_mv | AT caspersm sphericalfouriertransformsonlocallycompactquantumgelfandpairs |