Compact quantum hypergroups and their representations

Yu. A. Chapovsky^{1} L. I. Vainerman^{2}

1 Institute of Mathematics, Ukrainian National Academy of Sciences, ul. Tereshchinkivs’ka, 3, Kiev 262601, Ukraine

2 UMR 9994 Institut de Mathématiques de Jussieu, case 191, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05 France

Abstract

A compact quantum hypergroup is a unital C^{∗}-algebra equipped with a completely positive coassociative coproduct. The most important examples of such a structure are associated with double cosets of compact quantum groups in the sense of S. L. Woronowicz. We give a precise definition of a compact quantum hypergroup; prove existence and uniqueness of the Haar measure, establish orthogonality relations for matrix elements of irreducible corepresentations; construct a Peter–Weyl theory for irreducible corepresentations.

Introduction

Let G be a compact group and K be its subgroup with Haar measures μ_{G} and μ_{K} respectively. Then the algebra of all continuous functions on G that are bi-invariant with respect to translations by elements from K, C_{KK}(G), can be equipped with the coproduct

(δ f)(x, y) := ∫_{K} f(xky)dμ_{K} (f ∈ C_{KK}(G))

that plays an important role in the theory of zonal spherical functions (see, for example [8]) arising from irreducible representations of G. Such a structure is an example of what now is called a hypergroup (see [2, 9]). The situation becomes similar if one considers functions on a compact quantum group in the sense of S. L. Woronowicz [18, 19]. Here, for a pair of compact quantum groups (A_{1}, A_{2}) and a surjection π : A_{1} → A_{2}, we can also consider an algebra A of bi-invariant elements [3, 11] and define a coproduct on this algebra [5, 6, 14, 15]. In this case, we say that A is endowed with a hypergroup structure. A number of examples of the mentioned type can be found in the papers cited above and in [17]. All of these examples are associated with various classes of special and q-special functions.

1E-mail: yc@imath.kiev.ua

2E-mail: vaynerma@mpim-bonn.mpg.de, vainerma@math.jussieu.fr

Analyzing these examples, one can come to the idea of considering a general structure consisting of a unital C^{∗}-algebra equipped with a completely positive coassociative coproduct that preserves unit element and satisfies some additional natural axioms. Since compact quantum groups and usual hypergroups are included in this framework, we call such a structure a compact quantum hypergroup. The aim of the paper is to show that the theory of these objects is as rich as the theory of compact quantum groups [18, 19]. In particular, we prove existence and uniqueness of the Haar measure and some basic results concerning harmonic analysis on a compact quantum hypergroup.

The paper is organized as follows. In Section 1, we define a hypergroup structure on a C^{∗}-algebra and prove some elementary facts. In Section 2, we prove existence and uniqueness of the Haar measure. In Section 3, two examples of a hypergroup structure on a C^{∗}-algebra are considered, those of a compact quantum group itself and of a compact quantum homogeneous space arising from a pair of compact quantum groups (A_{1}, A_{2}) and an epimorphism π : A_{1} → A_{2}. In Section 4 we give a definition of a compact quantum hypergroup, prove that, in the case where the C^{∗}-algebra is commutative, the compact quantum hypergroup is just the usual hypercomplex system ([2]), and, finally, we give some results that are used in Section 5. In Section 5, we summarize elements of the theory of corepresentations and prove theorems of the Peter–Weyl theory, namely, we prove that irreducible corepresentations are finite dimensional and their matrix elements are total both in the C^{∗}- algebra and its L_{2} completion with respect to the corresponding norms.

1 Definition of a hypergroup structure on a C^{∗}-algebra

Let (A, ·, 1, ∗) be a separable unital C^{∗}-algebra. We denote by A ⊗ A the injective or projective C^{∗}-tensor square of A.

Definition 1.1 We will call (A, δ, ϵ, ⋆) a hypergroup structure on the C^{∗}- algebra (A, ·, 1, ∗) if:

(HS_{1}) (A, δ, ϵ, ⋆) is a ⋆-coalgebra with a counit ϵ, i.e. δ : A → A ⊗ A and ϵ : A → C are linear mappings, ⋆ : A → A is an antilinear mapping such that

(δ ⊗ id) ∘ δ = (id ⊗ δ) ∘ δ,  (1)
(ϵ ⊗ id) ∘ δ = (id ⊗ ϵ) ∘ δ = id,  (2)
δ ∘ ⋆ = Π ∘ (⋆ ⊗ ⋆) ∘ δ,  (3)
⋆ ∘ ⋆ = id,  (4)

where Π : A ⊗ A → A ⊗ A is the flip Π(a_{1} ⊗ a_{2}) = a_{2} ⊗ a_{1};

(HS_{2}) the mapping δ : A → A⊗A is positive, i.e. it maps the cone of positive elements of A into the cone of positive elements of A ⊗ A;

(HS_{3}) the following identities hold

(a · b)^{⋆} = a^{⋆} · b^{⋆},  δ ∘ ∗ = (∗ ⊗ ∗) ∘ δ,  (5)
ϵ(a · b) = ϵ(a)ϵ(b),  δ(1) = 1 ⊗ 1,  (6)
⋆ ∘ ∗ = ∗ ∘ ⋆.  (7)

Lemma 1.1 Let (A, δ, ϵ, ⋆) be a hypergroup structure on a unital C^{∗}-algebra (A, ·, 1, ∗). Then:

1^{⋆} = 1,  (8)
ϵ(1) = 1,  (9)
ϵ(a^{⋆}) = \overline{ϵ(a)},  (10)
ϵ(a^{∗}) = \overline{ϵ(a)}.  (11)

Proof. By using the first equality of (5), we have 1^{⋆} = (1 · 1)^{⋆} = 1^{⋆} · 1^{⋆}. Hence, 1^{⋆} · (1 − 1^{⋆}) = 0. And so, 1 · (1^{⋆} − 1) = 1^{⋆} − 1 = 0. This shows (8).

By using relations (2) and the second relation of (6), we have 1 = (ϵ ⊗ id) ∘ δ(1) = ϵ(1)1. Whence ϵ(1) = 1.

To show (10), let a ∈ A and consider (ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ⋆) ∘ δ(a^{⋆}). On one hand,

(ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ⋆) ∘ δ(a^{⋆}) = \overline{(\overline{ϵ} ⊗ ϵ) ∘ (id ⊗ ⋆) ∘ δ(a^{⋆})} =
= \overline{(ϵ ⊗ \overline{ϵ}) ∘ (⋆ ⊗ id) ∘ (⋆ ⊗ ⋆) ∘ δ(a)} =
= \overline{(ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ⋆) ∘ δ(a)} =
= \overline{\overline{ϵ}(a^{⋆})} = ϵ(a^{⋆}).

On the other hand,

(ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ⋆) ∘ δ(a^{⋆}) = \overline{ϵ}((a^{⋆})^{⋆})) = \overline{ϵ}(a).

Finally, (11) holds because of the following two identities:

(ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ∗) ∘ δ(a^{∗}) = \overline{(\overline{ϵ} ⊗ ϵ) ∘ (id ⊗ ∗) ∘ δ(a^{∗})} =
= \overline{(\overline{ϵ} ⊗ ϵ) ∘ (∗ ⊗ id) ∘ δ(a)} =
= \overline{\overline{ϵ}(a^{∗})} = ϵ(a^{∗})

and

(ϵ ⊗ \overline{ϵ}) ∘ (id ⊗ ∗) ∘ δ(a^{∗}) = \overline{ϵ}((a^{∗})^{∗}) = \overline{ϵ}(a).
□

By A^{∘} we denote the set of all continuous linear functionals on the C^{∗}- algebra A. For ξ, η ∈ A^{∘} we define a product · and an involution ^{+} by

\begin{matrix}(ξ · η)(a) = (ξ ⊗ η)δ(a) \\ ξ^{+}(a) = \overline{ξ(a^{⋆})}, \end{matrix}  (12)

a ∈ A, with the norm given by

∥ξ∥ = sup_{∥a∥=1} |ξ(a)|.  (13)

Lemma 1.2 Let the product, involution and the norm on A^{∘} be given by (12) and (13). Then (A^{∘}, ·, ϵ, ^{+}) is a unital Banach ∗-algebra.

Proof. It is clear that (A^{∘}, ·, ϵ, ^{+}) is a unital involutive algebra. Let us show that ∥ξ · η∥ ≤ ∥ξ∥ ∥η∥. Indeed, because δ : A → A ⊗ A is a positive mapping, ∥δ∥ = 1 [4, Corollary 3.2.6]. We also have for all ξ, η ∈ A^{∘} and ã ∈ A ⊗ A that (ξ ⊗ η)(ã) ≤ ∥ã∥ ∥ξ∥ ∥η∥ [13, Section IV.4]. Thus

∥ξ · η∥ = sup_{a∈A, ∥a∥=1} |(ξ ⊗ η)δ(a)| ≤ sup_{ã∈A⊗A, ∥ã∥=1} |(ξ ⊗ η)(ã)| ≤
≤ sup_{ã∈A⊗A, ∥ã∥=1} ∥ã∥ ∥ξ∥ ∥η∥ = ∥ξ∥ ∥η∥.

It remains to show that ∥ξ^{+}∥ = ∥ξ∥. First observe that ⋆ is an antilinear mapping, which preserves the cone of positive elements of the C^{∗}-algebra A. Indeed, it follows from (5) and (7) that

(a · a^{∗})^{⋆} = a^{⋆} · (a^{∗})^{⋆} = a^{⋆} · (a^{⋆})^{∗}.

Hence the mapping Φ : A → A defined by Φ(a) = (a^{∗})^{⋆} is a linear positive mapping. Thus ∥Φ∥ = 1. And so

∥ξ^{+}∥ = sup_{a∈A, ∥a∥=1} |\overline{ξ(a^{⋆})}| = sup_{a^{∗}∈A, ∥a^{∗}∥=1} |ξ((a^{∗})^{⋆})| =
= sup_{a∈A, ∥a∥=1} |ξ(Φ(a))| ≤ ∥ξ∥.

And, because + ∘ + = id, we have that ∥ξ∥ = ∥ξ^{++}∥ ≤ ∥ξ^{+}∥, whence ∥ξ^{+}∥ = ∥ξ∥.

It is clear that A^{∘} is a Banach space with respect to the norm (13).

□

2 Existence of a Haar measure

Definition 2.1 Let (A, δ, ϵ, ⋆) be a hypergroup structure on a C^{∗}-algebra A. A state ν ∈ A^{∘} is called a Haar measure (with respect to the hypergroup structure) if

(ν ⊗ id) ∘ δ(a) = (id ⊗ ν) ∘ δ(a) = ν(a)1  (14)

for all a ∈ A.

Definition 2.2 Let (A, δ, ϵ, ⋆) be a hypergroup structure on the C^{∗}-algebra A. An element a ∈ A is called positive definite if

ξ · ξ^{+}(a) ≥ 0  (15)

for all ξ ∈ A^{∘}.

To prove the following theorem, we combine the approaches of [10], [18], and [19].

Theorem 2.1 Let (A, δ, ϵ, ⋆) be a hypergroup structure on a C^{∗}-algebra A. Suppose that the linear space spanned by positive definite elements is dense in A. Then there exists a Haar measure ν, it is unique, and ν^{+} = ν.

Proof. Denote by Σ the set of all states on the C^{∗}-algebra A. Because δ is positive and δ(1) = 1 ⊗ 1, Σ is closed with respect to multiplication ·. Because ⋆ preserves the cone of positive elements, Σ is closed with respect to the involution ^{+}. Clearly, it is convex and compact in ∗-weak topology.

Let L = {Λ : Λ ⊂ Σ} denote the family of nonempty compact convex subsets of Σ such that Σ · Λ ⊂ Λ. The family L is nonvoid since Σ ∈ L. Consider L with partial ordering induced by inclusion. A standard argument using Zorn’s lemma shows that there is a minimal element Λ_{0} ∈ L.

For each λ ∈ Λ_{0}, Λ_{0} · λ = Λ_{0}. Indeed, Λ_{0} · λ ⊂ Σ · Λ_{0} ⊂ Λ_{0}. Moreover, Λ_{0} · λ ∈ L because Σ · (Λ_{0} · λ) = (Σ · Λ_{0}) · λ ⊂ Λ_{0} · λ. And, because Λ_{0} is minimal, Λ_{0} = Λ_{0} · λ. This implies that, for each λ, μ ∈ Λ_{0} there exists an element χ ∈ Λ_{0} such that

χ · λ = μ.  (16)

Denote by R = {Ξ : Ξ ⊂ Σ} the family of nonempty compact convex subsets of Σ such that Ξ·Σ ⊂ Ξ. Denote Λ_{0}^{+} = {λ^{+} : λ ∈ Λ_{0}}. Then, because Λ_{0}^{+} · Σ = (Σ · Λ_{0})^{+} ⊂ Λ_{0}^{+}, we see that Λ_{0}^{+} ∈ R. It is immediate that it is a minimal element of R. Hence, in the same way as before, for each ρ, σ ∈ Λ_{0}^{+}, there is an element ψ ∈ Λ_{0}^{+} such that

ρ · ψ = σ.  (17)

Now, let Ω = Λ_{0} ∩ Λ_{0}^{+}. The set Ω is nonempty because Λ_{0} ≠ ∅, Λ_{0}^{+} ≠ ∅ and Ω = Λ_{0} ⋂ Λ_{0}^{+} ⊃ Λ_{0}^{+} · Λ_{0}. Let ω ∈ Ω. Then it follows from (16) that there is ν_{ω}^{l} ∈ Λ_{0} such that

ν_{ω}^{l} · ω = ω.  (18)

By using (17) we see that, for all ρ ∈ Λ_{0}^{+} there is an element ψ_{ρ} ∈ Λ_{0}^{+} such that ω·ψ_{ρ} = ρ. This, together with (18) implies that, for an arbitrary ρ ∈ Λ_{0}^{+},

ν_{ω}^{l} · ρ = ρ.  (19)

Indeed,

ν_{ω}^{l} · ρ = ν_{ω}^{l} · (ω · ψ_{ρ}) = (ν_{ω}^{l} · ω) · ψ_{ρ} = ω · ψ_{ρ} = ρ.

By a similar argument we get ν_{ω}^{r} ∈ Λ_{0}^{+} such that

λ · ν_{ω}^{r} = λ  (20)

for all λ ∈ Λ_{0}. So (19) together with (20) mean that

ν_{ω}^{l} = ν_{ω}^{l} · ν_{ω}^{r} = ν_{ω}^{r}.

Denote by ν_{ω} = ν_{ω}^{l} = ν_{ω}^{r} ∈ Ω. If ν_{ω′} is another element in Ω verifying (19) and (20), then clearly ν_{ω′} = ν_{ω}. Hence such an element is unique and we denote it by ν. Relations (20) and (19) now become

λ · ν = λ,  ν · ρ = ρ  (21)

for all λ ∈ Λ_{0} and ρ ∈ Λ_{0}^{+}.

For each compact convex subset Υ of Σ and an arbitrary α ∈ Υ, m ∈ ℤ_{+}, we denote

α^{(m)} = \frac{1}{m} ∑_{i=1}^{m} α^{i}.

Because Υ is compact, there is an accumulation point α^{(∞)} of the sequence α^{(m)} in Υ and, if necessary, considering a subsequence we can assume that α^{(∞)} = lim_{m→∞} α^{(m)}. Now

α · α^{(m)} = \frac{1}{m} ∑_{i=2}^{m+1} α^{m} = \frac{1}{m} ∑_{i=1}^{m} α^{i} + \frac{1}{m}(α^{m+1} − α) = α^{(m)} + \frac{1}{m}(α^{m+1} − α).

Since · is continuous, we get

lim_{m→∞}(α · α^{(m)}) = α · lim_{m→∞} α^{(m)} = α · α^{(∞)}.

On the other hand, because Υ is compact, it is bounded and hence we have that lim_{m→∞} \frac{1}{m}(α^{m+1} − α) = 0. This shows that

lim_{m→∞} \{α^{(m)} + \frac{1}{m}(α^{(m+1)} − α)\} = α^{(∞)}.

And so

α · α^{(∞)} = α^{(∞)}.  (22)

In the same way we get

α^{(∞)} · α = α^{(∞)}.  (23)

Now let λ ∈ Λ_{0}. For λ^{(∞)} ∈ Λ_{0} and ν ∈ Ω by using (16), we can find χ ∈ Λ_{0} such that

χ · λ^{(∞)} = ν.

But this means that

ν = χ · λ^{(∞)} = χ · λ^{(∞)} · λ = ν · λ.

In particular, for any ω ∈ Ω,

ν = ν · ω = ω.

This shows that Ω = Λ_{0}^{+} · Λ_{0} = {ν}, i.e.

ρ · λ = ν  (24)

for all λ ∈ Λ_{0}, ρ ∈ Λ_{0}^{+}.

Now choose an arbitrary element λ ∈ Λ_{0}. We will prove that λ = ν. Let Λ̃ be the ∗-algebra generated by the elements λ, λ^{+}, ν, ϵ. Denote ζ = ν − λ. It follows from (24) that

ζ^{+}·ζ = (ν −λ^{+})·(ν −λ) = ν ·ν −ν ·λ−λ^{+}·ν +λ^{+}·λ = ν −ν −ν +ν = 0. (25)

If p ∈ A is a positive definite element, then, considered as a linear functional on Λ̃, it is a positive linear functional, and, hence,

|(ν − λ)(p)| ≤ ϵ(p)((ν − λ)^{+} · (ν − λ))(p) = 0.

This means that ν(p) = λ(p) for any positive definite p ∈ A and, since each element in A can be approximated by a linear combination of positive definite elements, ν(a) = λ(a) for all a ∈ A, hence ν = λ.

So, we get that Λ = Λ^{+} = {ν}, which means that ν is a Haar measure, it is clearly unique, and by construction, ν^{+} = ν.

□

Remark 2.1 In defining a hypergroup structure on a C^{∗}-algebra, it would be natural to follow the lines taken in [19] for defining a compact quantum group. In particulr, if the following axiom, used in [19], is assumed

(W): the linear subsets ∑_{i=1}^{n}(b_{i} ⊗ 1)δ(a_{i}) and ∑_{i=1}^{n}(1 ⊗ b_{i})δ(a_{i}), a_{i}, b_{i} ∈ A, n ∈ ℤ_{+}, are dense in A ⊗ A,

then the claim in Theorem 2.1 would still be true and could be proved by using a slight modification of the proof found in [19]. Unfortunately, condition (W) is too strong as an assumption, which can be seen from the following example, and thus we replaced it, in the statement of Theorem 2.1, by the condition that positive definite elements are total in A.

Example 2.1 Let A = {f ∈ C(ℝ)|f(x) = f(x + 2π), f(x) = f(−x)} be a commutative C^{∗}-algebra of continuous real valued periodic even functions on ℝ. Let δ : A → A ⊗ A be given by δ(f)(x, y) = 1/2(f(x + y) + f(x − y)), ϵ(f) = f(0), f^{⋆}(x) = f(x). For a cocommutative δ, c ∈ A is called a character if δ(c) = c ⊗ c. In the case under consideration, the characters are c_{n} = cos nx, n ∈ ℤ_{+}. By the Weierstrass theorem, the linear span of the set C = {c_{n}|n ∈ ℤ_{+}} is dense in A but an easy argument shows that no element of the linear span of the set {δ(c_{k})(c_{l} ⊗ 1)|k, l ∈ ℤ_{+}} approximates, for example, the element c_{m} ⊗ c_{n} for m < n. Thus condition (W) is violated. However, the characters are positive definite functions and, hence, the density condition of Theorem 2.1 holds and the Haar measure exists and is given by ν(f) = 1/π ∫_{0}^{π} f(x) dx.

3 Examples

3.1 A hypergroup structure associated with a compact quantum group

Let (A, ·, 1, ∗, Δ, ϵ, S) be a compact quantum group with A_{0} being the involutive subalgebra generated by matrix elements of the fundamental corepresentation [18, 19]. We also use the following notations

ξ . a = (id⊗ξ)∘Δ(a),  a . ξ = (ξ ⊗id)∘Δ(a),  ξ ·η = (ξ ⊗η)∘Δ (26)

for ξ, η ∈ A^{∘} and a ∈ A, and

Δ^{(1)} = Δ,  Δ^{(2)} = (id ⊗ Δ) ∘ Δ,  etc.

It readily follows from (26) that

ξ . (η . a) = (ξ · η) . a,  (a . η) . ξ = a . (η · ξ).  (27)

Let U^{α} = (u_{ij}^{α})_{i,j=1}^{d_{α}} be an irreducible unitary corepresentation of A. Then there exists a unique, up to a positive constant, positive definite matrix M^{α} = (m_{ij}^{α})_{i,j=1}^{d_{α}} such that

M^{α} · U^{α} = S^{2}(U^{α}) · M^{α},  (28)

where · here denotes the usual matrix multiplication [18].

For each z ∈ ℂ, we denote by m_{ij}^{α(z)} the matrix elements of the matrix (M^{α})^{z}. It is known that there exists a one-parameter family of homomorphisms f_{z} : A_{0} → ℂ, z ∈ ℂ, where, as before, A_{0} denotes the ∗-subalgebra generated by matrix elements of the fundamental corepresentation. These homomorphisms are defined by

f_{z}(u_{ij}^{α}) = m_{ij}^{α(z)}  (29)

and possess the following properties [18]:

(F_{1}): f_{z}(1) = 1 for all z ∈ ℂ;

(F_{2}): f_{z} · f_{z′} = f_{z+z′} and f_{0} = ϵ; (F_{3}): f_{z}(S(a)) = f_{−z}(a);

(F_{4}): f_{z}(a^{∗}) = \overline{f_{−\overline{z}}(a)};

(F_{5}): S^{2}(a) = f_{−1} . a . f_{1};

(F_{6}): ν(a · b) = ν(b · (f_{1} . a . f_{1})), where ν is the Haar measure on the compact quantum group A.

Now define a mapping ⋆ : A_{0} → A_{0} by

a^{⋆} = f_{−1/2} . S(a)^{∗} . f_{1/2}.  (30)

Lemma 3.1 Let the mapping ⋆ be defined by (30). Then (A_{0}, Δ, ϵ, ⋆) is an involutive coalgebra. Moreover the first relation of (5) and relation (7) hold. The mapping ⋆ is continuous.

Proof. It is clear from definition (30) that ⋆ is an antilinear and multiplicative mapping.

Let us show that Δ ∘ ⋆ = (⋆ ⊗ ⋆) ∘ Δ. Indeed, by using property (F_{2}) and definition (30), we have

Δ ∘ ⋆ = Δ ∘ (f_{−1/2} ⊗ ∗ ∘ S ⊗ f_{1/2}) ∘ Δ^{(2)} =
= Π ∘ (f_{−1/2} ⊗ ∗ ∘ S ⊗ ∗ ∘ S ⊗ f_{1/2}) ∘ Δ^{(3)} =
= Π ∘ (f_{−1/2} ⊗ ∗ ∘ S ⊗ f_{1/2} ⊗ f_{−1/2} ⊗ ∗ ∘ S ⊗ f_{1/2}) ∘ Δ^{(5)} =
= Π ∘ (⋆ ⊗ ⋆) ∘ Δ.

To show that ⋆ ∘ ⋆ = id, first note that, by property (F_{3}), we have that, for all z, z′ ∈ ℂ,

f_{z} . S(a) . f_{z′} = (f_{z′} ⊗ id ⊗ f_{z}) ∘ Δ^{(2)}(a) =
= (f_{z} ∘ S ⊗ S ⊗ f_{z′} ∘ S) ∘ Δ^{(2)}(a) =
= (f_{−z} ⊗ S ⊗ f_{−z′}) ∘ Δ^{(2)} =
= S(f_{−z′} . a . f_{−z}),  (31)

and, by property (F_{4}),

f_{z} . a^{∗} . f_{z′} = (f_{z′} ⊗ id ⊗ f_{z}) ∘ Δ^{(2)}(a^{∗}) =
= (f_{z′} ∘ ∗ ⊗ ∗ ⊗ f_{z} ∘ ∗) ∘ Δ^{(2)} =
= (\overline{f}_{−\overline{z}′} ⊗ ∗ ⊗ \overline{f}_{−\overline{z}}) ∘ Δ^{(2)} =
= (f_{−\overline{z}} . a . f_{−\overline{z}′})^{∗}.  (32)

Now, by using (31) and (32), we obtain:

a^{⋆} = f_{−1/2} . S(a)^{∗} . f_{1/2} = (f_{1/2} . S(a) . f_{−1/2})^{∗} =
= S(f_{1/2} . a . f_{−1/2})^{∗}.  (33)

This implies that

(a^{⋆})^{⋆} = f_{−1/2} . S(S(f_{1/2} . a . f_{−1/2})^{∗})^{∗} . f_{1/2} = f_{−1/2} . (f_{1/2} . a . f_{−1/2}) . f_{1/2} =
= (f_{−1/2} · f_{1/2}) . a . (f_{−1/2} · f_{1/2}) = a.

We now show that ∗ and ⋆ commute. Indeed,

S((a^{⋆})^{∗}) = S((f_{−1/2} . S(a)^{∗} . f_{1/2})^{∗}) =
= S(f_{1/2}) . S(a) . f_{−1/2}) =
= f_{1/2} . .S^{2}(s) . f_{−1/2} =
= (f_{1/2} · f_{−1}) . a . (f_{1} · f_{−1/2}) =
= f_{−1/2} . a . f_{1/2}.

On the other hand,

S((a^{∗})^{⋆}) = S(f_{−1/2} . S(a^{∗})^{∗} . f_{1/2}) =
= f_{−1/2} . S(S(a^{∗})^{∗}) . f_{1/2} =
= f_{−1/2} . a . f_{1/2}.

Because the antipode S is invertible, we see that ⋆ ∘ ∗ = ∗ ∘ ⋆.

Continuity of ⋆ is a direct consequence of its commutativity with involution.

□

Remark 3.1 Because ⋆ is continuous and A_{0} is dense in A, we can extend ⋆ by continuity to the whole C^{∗}-algebra A and consider the coinvolution to be defined on A.

Corollary 3.1 Let (A, ·, 1, ∗, Δ, ϵ, S) be a compact quantum group in the sense of [18, 19]. Then (A, Δ, ϵ, ⋆) is a hypergroup structure on the C^{∗}- algebra (A, ·, 1, ∗).

3.2 A hypergroup structure associated with a quantum homogeneous space

Let now (A_{1}, ·, 1, ∗, Δ_{1}, ϵ_{1}, S_{1}) and (A_{2}, ·, 1, ∗, Δ_{2}, ϵ_{2}, S_{2}) be two compact quantum groups and let π : A_{1} → A_{2} be a Hopf C^{∗}-algebra epimorphism, i.e. π is a C^{∗}-algebra epimorphism satisfying (π ⊗ π) ∘ Δ_{1} = Δ_{2} ∘ π, ϵ_{2} ∘ π = ϵ_{1} on A, and also π(A_{1}_{0}) ⊂ A_{2}_{0} with π ∘ S_{1} = S_{2} ∘ π on A_{1}_{0}, where A_{i}_{0} is the ∗-subalgebra generated by matrix elements of the fundamental corepresentation of the corresponding quantum group, i = 1, 2.

Lemma 3.2 If π : A_{1} → A_{2} is a Hopf ∗-algebra epimorphism and the coinvolution ⋆ is defined by (30) on each A_{i}, i = 1, 2, then

⋆ ∘ π = π ∘ ⋆.  (34)

Proof. Consider a basis in the space of corepresentation of A_{1} such that, for the matrix elements U^{α} in this basis, we have

π(U^{α}) = ( \begin{matrix}V^{β} & 0 \\ 0 & I \end{matrix} ) ,  (35)

where V^{β} is an irreducible unitary corepresentation of A_{2}. Write the positive definite matrix M^{α} from (29) as

M^{α} = ( \begin{matrix}M_{11}^{α} & M_{12}^{α} \\ M_{21}^{α} & M_{22}^{α} \end{matrix} ) .

Since π commutes with S, applying π to (28) and using the form (35), we see that

( \begin{matrix}M_{11}^{α} & M_{12}^{α} \\ M_{21}^{α} & M_{22}^{α} \end{matrix} ) ( \begin{matrix}V^{β} & 0 \\ 0 & I \end{matrix} ) = ( \begin{matrix}S^{2}(V^{β}) & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}M_{11}^{α} & M_{12}^{α} \\ M_{21}^{α} & M_{22}^{α} \end{matrix} ) .

This implies that

\begin{matrix}M_{11}^{α}V^{β} = S^{2}(V^{β})M_{11}^{α}, \\ M_{12}^{α} = M_{12}^{α}S^{2}(V^{β}), M_{21}^{α}V^{β} = M_{21}^{α}. \end{matrix}

The first identity means, because the matrix M^{α} is unique up to a constant, that M^{β} = k M_{11}^{α} for some positive k, and the second and third — that M_{12}^{α} = M_{21}^{α} = 0. Now, if we write (30) as

(U^{α})^{⋆} = (M^{α})^{1/2}S(U^{α})^{∗}(M^{α})^{−1/2},

and use that π commutes with S and ∗, we get that

π((U^{α})^{⋆}) = ( \begin{matrix}M_{11}^{α} & 0 \\ 0 & M_{22}^{α} \end{matrix} )^{1/2} ( \begin{matrix}S(V^{β})^{∗} & 0 \\ 0 & I \end{matrix} ) ( \begin{matrix}M_{11}^{α} & 0 \\ 0 & M_{22}^{α} \end{matrix} )^{−1/2} =
= ( \begin{matrix}(M_{11}^{α})^{1/2}S(V^{β})^{∗}(M_{11}^{α})^{−1/2} & 0 \\ 0 & I \end{matrix} ) =
= ( \begin{matrix}(M^{β})^{1/2}S(V^{β})^{∗}(M^{β})^{−1/2} & 0 \\ 0 & I \end{matrix} ) =
= ( \begin{matrix}(V^{β})^{⋆} & 0 \\ 0 & I \end{matrix} ) = π(V^{α})^{⋆}.
□

Consider now two compact quantum groups (A_{i}, ·, 1, ∗, Δ_{i}, ϵ_{i}, S_{i}), i = 1, 2, and let π : A_{1} → A_{2} be a Hopf ∗-algebra epimorphism. Define

\begin{matrix}A_{1}/A_{2} = \{a ∈ A_{1} : (id ⊗ π) ∘ δ(a) = a ⊗ 1\}, \\ A_{2}\A_{1} = \{a ∈ A_{1} : (π ⊗ id) ∘ δ(a) = 1 ⊗ a\}, \\ A_{2}\A_{1}/A_{2} = A_{2}\A_{1} ⋂ A_{1}/A_{2}. \end{matrix}  (36)

It is immediate that A_{1}/A_{2}, A_{2}\A_{1}, A_{2}\A_{1}/A_{2} are involutive algebras with the unit 1.

Following [5, 6], we define δ : A_{1} → A_{1} ⊗ A_{1} by

δ = (id ⊗ ν_{2} ∘ π ⊗ id) ∘ Δ_{1}^{(2)},  (37)

and let ⋆ : A_{1} → A_{1} be given by

a^{⋆} = f_{−1/2} · S_{1}(a)^{∗} · f_{1/2}.  (38)

Here ν_{2} is the Haar measure for A_{2} and f_{z}, z ∈ ℂ, is the one-parameter family of modular homomorphisms on A_{1}.

It is easy to show that δ : A_{2}\A_{1}/A_{2} → A_{2}\A_{1}/A_{2} ⊗ A_{2}\A_{1}/A_{2} and, since π and ⋆ commute, ⋆ : A_{2}\A_{1}/A_{2} → A_{2}\A_{1}/A_{2}. Denote by A^{inv} the C^{∗}-algebra completion of A_{2}\A_{1}/A_{2} and extend the mappings δ and ⋆ by continuity to the corresponding mappings on A^{inv}.

Theorem 3.1 Let (A_{i}, ·, 1, ∗, Δ_{i}, ϵ_{1}, S_{i}), i = 1, 2, be two compact quantum groups and π : A_{1} → A_{2} — a Hopf *-algebra epimorphism. Then, with the mappings δ, ^{⋆} defined by (37) and (38), and ϵ = ϵ_{1}|_{A^{inv}}, (A^{inv}, δ, ϵ, ⋆) is a hypergroup structure on the C^{∗}-algebra (A^{inv}, ·, 1, ∗).

Proof. (HS_{1}): Identity (4) has already been proved in Lemma 3.1. To see that (3) holds, first recall that (A_{2}, Δ_{2}, ϵ, ⋆) is a quantum hypergroup structure on the C^{∗}-algebra (A, ·, 1, ∗) and ν_{2} is the Haar measure since it is unique. Moreover ν_{2}^{+} = \overline{ν}_{2} ∘ ⋆ = ν_{2}, whence ν_{2} ∘ ⋆ = \overline{ν}_{2}. Because Δ_{1} ∘ ⋆ = Π ∘ (⋆ ⊗ ⋆) ∘ Δ_{1} (see the proof of Lemma 3.1) and ⋆ and π commute, we get (3). The proof of (1) and (2) is easy.

(HS_{2}): The mapping δ is a composition of the homomorphism Δ_{1}^{(2)}, which is positive, and ν_{2}, which is a positive functional on a C^{∗}-algebra. Hence δ is positive.

All properties in (HS_{3}) are immediate.

□

4 Compact quantum hypergroup

Definition 4.1 Suppose that (A, δ, ϵ, ⋆) is a hypergroup structure on a C^{∗}- algebra (A, ·, 1, ∗). We call A = (A, ·, 1, ∗, δ, ϵ, ⋆, σ_{t}) a compact quantum hypergroup if

(QH_{1}) the mapping δ is completely positive [13] and the linear span of positive definite elements is dense in A;

(QH_{2}) σ_{t}, t ∈ ℝ, is a continuous one-parameter group of automorphisms of A such that

(a) the set A_{0} of analytic elements of A, i.e. the set of elements a ∈ A such that the function t ↦ σ_{t}(a) ∈ A can be extended to an analytic function on ℂ, is dense in A;

(b) A_{0} is invariant with respect to ∗ and ⋆, and δ(A_{0}) ⊂ A_{0} ⊗ A_{0};

(c) the following relations hold on A_{0} for all z ∈ ℂ:

δ ∘ σ_{z} = (σ_{z} ⊗ σ_{z}) ∘ δ,  (39)
ν(σ_{z}(a)) = ν(a);  (40)

(d) there exists z_{0} ∈ ℂ such that the Haar measure ν satisfies the following strong invariance condition for all a, b ∈ A_{0}:

(id⊗ν)[((∗∘σ_{z_{0}}∘⋆⊗id)∘δ(a))·(1⊗b)] = (id⊗ν)((1⊗a)·δ(b)); (41)

(QH_{3}) the Haar measure ν is faithful on A_{0}.

In the sequel it will be convenient to denote

κ = ∗ ∘ σ_{z_{0}} ∘ ⋆  (42)

and call it an antipode. Note that κ is invertible with κ^{−1} = ⋆ ∘ σ_{−z_{0}} ∘ ∗. With such a notation, relation (41) becomes

(id ⊗ ν)((κ ⊗ id) ∘ δ(a) · (1 ⊗ b)) = (id ⊗ ν)((1 ⊗ a) · δ(b)).  (43)

Remark 4.1 The definition of a compact quantum hypergroup involves the use of a continuous one-parameter group of automorphisms. At this point, we followed the approach used to define a Woronowicz algebra [12]. If the group of automorphisms acts trivially, then this is a way a Kac algebra is defined (see [7, 16]).

Example 4.1 The examples of hypergroup structures considered in Section 3 are, in fact, examples of compact quantum hypergroups. Consider, for instance, the example of (A^{inv}, δ, ϵ, ⋆), a hypergroup structure associated with a quantum homogeneous space. Here A_{0} = A^{inv} ⋂ A_{1,0}, where A_{1,0} is the ∗-subalgebra of A_{1} generated by matrix elements of the fundamental corepresentation. The action of the group σ_{t} is defined by σ_{t}(a) = f_{it} . a . f_{−it}, z_{0} = −\frac{1}{2}i, and κ = S.

In this case δ is completely positive since it is a composition of completely positive maps. Axioms (QH_{2}) and (QH_{3}) summarize well known properties of compact quantum groups [18].

Lemma 4.1 Let A be a compact quantum hypergroup and κ be defined by (42). Then

\begin{matrix}κ(ab) = κ(b)κ(a),  δ ∘ κ = Π ∘ (κ ⊗ κ) ∘ δ, \\ ν ∘ κ = ν,  κ(1) = 1,  ϵ ∘ κ = ϵ. \end{matrix}  (44)

Proof. The first four identities are obvious. To show the last identity, let us show that ϵ(σ_{z}(a)) = ϵ(a), a ∈ A_{0}, z ∈ ℂ. By using (39), we have

ϵ(σ_{z}(a)) = ϵ(σ_{z} ∘ (ϵ ⊗ id) ∘ δ(a))
= (ϵ ⊗ ϵ) ∘ (id ⊗ σ_{z}) ∘ δ(a)
= (ϵ ⊗ ϵ) ∘ (σ_{−z} ⊗ id) ∘ (σ_{z} ⊗ σ_{z}) ∘ δ(a)
= (ϵ ⊗ ϵ) ∘ (σ_{−z} ⊗ id) ∘ δ(σ_{z}(a)) = ϵ(σ_{−z}(σ_{z}(a)))
= ϵ(a).

□

If A is a compact quantum hypergroup, we use the GNS construction to complete A_{0} or A to a Hilbert space H_{ν} with respect to the norm ∥ · ∥_{ν} induced by the inner product

⟨a, b⟩ = ν(b^{∗}a).  (45)

Proposition 4.1 Let A be a compact quantum hypergroup and, for a ∈ A_{0}, let an operator T_{a} : A_{0} → A_{0} be defined by

T_{a}(x) = (id ⊗ ν)((κ ⊗ id) ∘ δ(a) · (1 ⊗ x)) = (id ⊗ ν)((1 ⊗ a) · δ(x)). (46)

Then

(a) the operator T_{a}, a ∈ A_{0}, is a Hilbert-Schmidt type operator if extended by continuity to the operator T_{a} : H_{ν} → H_{ν};

(b) for x ∈ H_{ν}, a ∈ A_{0}, the following relation holds

∥T_{a}(x)∥ ≤ ∥a∥ ∥x∥_{ν}  (47)

and, hence, for a ∈ A_{0}, the range Ran(T_{a}) ⊂ A;

(c) the adjoint operator is given by

T_{a}^{†} = (ν ⊗ id)[((∗ ∘ κ ∘ ∗ ⊗ id) ∘ δ(a^{∗})) · (x ⊗ 1)];  (48)

(d) denote by R the set {T_{a}(b) : a, b ∈ A_{0}}. Then R is total in H_{ν} with respect to the norm ∥ · ∥_{ν}.

Proof. (a). Let e_{i} ∈ A_{0}, i ∈ ℤ_{+}, be an orthonormal basis in H_{ν}, and denote ã = (κ ⊗ id) ∘ δ(a) ∈ A_{0} ⊗ A_{0}. Then T_{a}(x) = (id ⊗ ν)(ã · (1 ⊗ x)). We have

∑_{i,j∈ℤ_{+}}⟨T_{a}(e_{i}), e_{j}⟩^{2} = ∑_{i,j∈ℤ_{+}} ν(e_{j}^{∗} · T_{a}(e_{i}))^{2}
= ∑_{i,j∈ℤ_{+}} ν(e_{j}^{∗} · (id ⊗ ν)(ã · (1 ⊗ e_{i})))^{2}
= ∑_{i,j∈ℤ_{+}}(\overline{ν} ⊗ ν)((∗ ⊗ id)(ã) · (e_{j} ⊗ e_{i}))^{2}.

The last sum is finite since (\overline{ν}⊗ν)((∗⊗id)(ã)·(e_{j}⊗e_{i})) is the Fourier coefficient of the element (id ⊗ ∗)(ã) considered in the Hilbert space \overline{H}_{ν} ⊗ H_{ν}, where the Hilbert space \overline{H}_{ν} is obtained by completing A_{0} with respect to the inner product ⟨a, b⟩ν̄ = \overline{ν}(b^{∗}a).

(b). Let φ be a state on the C^{∗}-algebra A, a ∈ A_{0}, x ∈ H. Then we have that

φ(T_{a}(x)^{∗} · T_{a}(x)) = φ((id ⊗ ν)((1 ⊗ a) · δ(x))^{∗} · (id ⊗ ν)((1 ⊗ a) · δ(x)))
≤ φ((id ⊗ ν)(δ(x)^{∗} · (1 ⊗ a^{∗}) · (1 ⊗ a) · δ(x)))
= (φ ⊗ ν)(δ(x)^{∗} · (1 ⊗ a^{∗}a) · δ(x))
≤ (φ ⊗ ν)(δ(x)^{∗} · δ(x)) ∥1 ⊗ a∥^{2} ≤ (φ ⊗ ν)(δ(x^{∗}x)) ∥a∥^{2}
= ν(x^{∗}x)∥a∥^{2} = ∥x∥_{ν}^{2} ∥a∥^{2}.

Here we used the fact that the mappings (id⊗ν) and δ are completely positive and that ν is a Haar measure. This proves (47).

Finally, to show that T_{a}(x) ∈ A, let x_{n} →_{ν} x with x_{n} ∈ A_{0}. All T_{a}(x_{n}) ∈ A_{0} and relation (47) shows that T_{a}(x_{n}) → T_{a}(x) with respect to the C^{∗}-norm. Since A is closed, the claim follows.

(c). To prove formula (48), we have

⟨T_{a}(x), y⟩ = ν(y^{∗} · T_{a}(x))
= ν(y^{∗} · (id ⊗ ν)((κ ⊗ id) ∘ δ(a) · (1 ⊗ x)))
= (ν ⊗ ν)((y^{∗} ⊗ 1) · (κ ⊗ id) ∘ δ(a) · (1 ⊗ x))
= ν(((ν ⊗ id)((y^{∗} ⊗ 1) · (κ ⊗ id) ∘ δ(a)) · x).

Hence

T_{a}^{†}(y) = (ν ⊗ id)((y^{∗} ⊗ 1) · (κ ⊗ id) ∘ δ(a))^{∗}
= (ν ⊗ id)((∗ ∘ κ ⊗ ∗) ∘ δ(a) · (y ⊗ 1))
= (ν ⊗ id)((∗ ∘ κ ∘ ∗ ⊗ id) ∘ δ(a^{∗}) · (y ⊗ 1)).

(d). Suppose that R is not total. Then there exists an element y ∈ H_{ν}, y ≠ 0, such that y ⊥ T_{a}(b), b ∈ A_{0}, or, which is the same thing, that for all a ∈ A_{0},

T_{a}^{†}(y) = (ν ⊗ id)((∗ ∘ κ ∘ ∗ ⊗ id) ∘ δ(a^{∗}) · (y ⊗ 1)) = 0.

By taking ϵ of both sides of this equality, we obtain that ν(κ(a)^{∗}y) = 0. By taking a = κ^{−1}(c^{∗}), we see that ν(cy) = 0 for all c ∈ A_{0}. But this contradicts density of A_{0} in H_{ν}.

□

Lemma 4.2 Let A be a compact quantum hypergroup. Then

∗ ∘ κ ∘ ∗ ∘ κ = id  (49)

on A_{0} and the operator T_{a}^{†}, given by (48), can be written in the form

T_{a}^{†} = T_{a^{†}},  (50)

where a^{†} on A_{0} is defined by

a^{†} = κ(a)^{∗}.  (51)

Proof. Let a, b ∈ A_{0}. Rewrite the identity (43)

(id ⊗ ν)((κ ⊗ id) ∘ δ(a) · (1 ⊗ b)) = (id ⊗ ν)((1 ⊗ a) · δ(b))

and take ∗ of both sides to get

(id ⊗ \overline{ν})((∗ ∘ κ ⊗ id) ∘ δ(a) · (1 ⊗ b)) = (id ⊗ ν)(δ(b^{∗}) · (1 ⊗ a^{∗})).

Now, by taking κ of both sides and using again (43), we obtain

(id ⊗ \overline{ν})((κ ∘ ∗ ∘ κ ⊗ id) ∘ δ(a) · (1 ⊗ b))

= (id ⊗ ν)((κ ⊗ id) ∘ δ(b^{∗}) · (1 ⊗ a^{∗}))

= (id ⊗ ν)((1 ⊗ b^{∗}) · δ(a^{∗})).

Now again apply ∗ to both sides:

(id ⊗ ν)((∗ ∘ κ ∘ ∗ ∘ κ ⊗ id) ∘ δ(a) · (1 ⊗ b)) = (id ⊗ ν)((δ(a) · (1 ⊗ b)).

This means that on the element T_{a}(b) ∈ R, ∗ ∘ κ ∗ ∘κ = id. But R is dense in H_{ν}, and hence, in A_{0}. Thus (49) holds.

Formula (50) follows immediately from (48) by noticing that (49) implies the relation ∗ ∘ κ = κ^{−1} ∘ ∗.

□

Formulas (49) and (51) show that, for all a ∈ A_{0}, a^{††} = a. This means that any element in A_{0} can be written as a = a_{r}+i a_{im} with a_{r}^{†} = a_{r}, a_{im}^{†} = a_{im} by setting a_{r} = \frac{1}{2}(a + a^{†}), a_{im} = \frac{1}{2i}(a − a^{†}). It also follows from (50) and part (a) of Proposition 4.1 that T_{a} is a compact self adjoint operator if a^{†} = a.

Proposition 4.2 The set R′ = {T_{a}(b) : a, b ∈ A_{0}, a^{†} = a} is total in A with respect to the C^{∗}-norm.

Proof. Suppose that the closure \overline{R′} ≠ A. Then there exists a ∈ A_{0} \ \overline{R′} and a continuous linear functional φ such that φ(R′) = 0 and φ(a) = ∥a∥. Let us consider the element b = (φ⊗id)∘δ(a). Because ϵ(b) = (φ⊗ϵ)∘δ(a) = φ(a) ≠ 0, we see that b ≠ 0. This means that ν(b^{∗}b) ≠ 0 since ν is faithful. On the other hand,

ν(b^{∗}b) = ν(b^{∗} · (φ ⊗ id) ∘ δ(a))
= (φ ⊗ ν)((1 ⊗ b^{∗}) · δ(a))
= φ(T_{b^{∗}}(a))
= φ(T_{b_{r}^{∗}}(a) + i T_{b_{im}^{∗}}(a)) = 0.

This contradiction proves the claim.

□

Suppose now that the C^{∗} algebra A is commutative and denote its spectrum by P. Each element ξ ∈ P defines a linear operator on A defined by

R_{ξ} = (id ⊗ ξ) ∘ δ,  (52)

which will be called a generalized translation operator. For ξ ∈ P and a ∈ A_{0}, we define

ξ^{†}(a) = \overline{ξ(a^{†})}.  (53)

From the definition of a^{†} it immediately follows that ξ^{†} is a homomorphism A_{0} → ℂ and, hence, continuous. Being extended by continuity to A, it becomes a point in P.

Lemma 4.3 Let A be a commutative compact quantum hypergroup. Let R_{ξ}, ξ ∈ P, be given by (52). Then:

(a) R_{ξ} is a bounded operator and the mapping ξ ↦ R_{ξ} is strongly continuous;

(b) for all ξ, η ∈ P and a ∈ A_{0}, \overline{ξ^{†}(R_{η^{†}}(a^{†}))} = η(R_{ξ}(a));

(c) the counit ϵ belongs to P and R_{ϵ} = id; (d) if a ∈ A is positive, then R_{ξ}(a) is also positive for all ξ ∈ P;

(e) for all ξ, η ∈ P, η(R_{ξ}(1)) = 1;

(f) if R_{ξ}^{†} denotes the operator adjoint to R_{ξ} with respect to inner product (45), then R_{ξ}^{†} = R_{ξ^{†}}.

Proof. (a) Because ξ is a homomorphism of a C^{∗} algebra, it is completely positive, and, hence, R_{ξ} is completely positive since δ is. This implies that R_{ξ}(a)^{∗} · R_{ξ}(a) ≤ R_{ξ}(a^{∗} · a) [13]. So, we have

∥R_{ξ}(a)∥_{ν}^{2} = ν(R_{ξ}(a)^{∗} · R_{ξ}(a))
≤ ν(R_{ξ}(a^{∗}a)) = (ξ ⊗ ν) ∘ δ(a^{∗}a)
= (ξ ⊗ id) ∘ (id ⊗ ν) ∘ δ(a^{∗} · a)
= ξ(ν(a^{∗}a)1) = ν(a^{∗}a)ξ(1)
= ∥a∥_{ν}^{2} ξ(1).

This shows that the operator R_{ξ} is bounded. It also follows that, if ξ → η, then ∥(R_{ξ} − R_{η})(a)∥_{ν} → 0, which means that R_{ξ} is strongly continuous.

(b) Let ξ, η ∈ P. By using (52) and (12), we have

\overline{ξ^{†}(R_{η^{†}}(a^{†}))} = \overline{(ξ^{†} ⊗ η^{†})} ∘ δ(a^{†}) = (ξ ⊗ η) ∘ († ⊗ †) ∘ δ(a^{†}) =
= (η ⊗ ξ) ∘ δ(a) = η(R_{ξ}(a)).

(c) The counit ϵ is a homomorphism and, hence ϵ ∈ P. Also R_{ϵ} = id.

(d) Because δ is positive by definition of a quantum hypergroup and ξ is positive as a homomorphism, L_{ξ} is positive.

(e) Because ξ and η are homomorphisms, this immediately follows from (6).

(f) Let a, b ∈ A_{0} and ξ ∈ P. Then

⟨R_{ξ}(a), b⟩ = ν(b^{∗} · R_{ξ}(a))
= ν(b^{∗} · (id ⊗ ξ) ∘ δ(a))
= (ν ⊗ ξ)((b^{∗} ⊗ 1) · δ(a)).

On the other hand, by using the definition of †, the fact that δ ∘ κ^{−1} = Π ∘ (κ^{−1} ⊗ κ^{−1}) ∘ δ and that κ is an antihomomorphism, as well as that ν ∘ κ = ν and (43), we obtain

⟨a, R_{ξ^{†}}(b)⟩ = ν(R_{ξ^{†}}(b)^{∗} · a) = ν((id ⊗ ξ^{†}) ∘ δ(b)^{∗} · a)
= ν((∗ ⊗ ξ ∘ †) ∘ δ(b) · a)
= (ν ⊗ ξ)((∗ ⊗ ∗ ∘ κ) ∘ δ(b) · (a ⊗ 1))
= (ν ⊗ ξ)((∗ ⊗ κ^{−1} ∘ ∗) ∘ δ(a) · (a ⊗ 1))
= (ν ⊗ ξ)((κ ∘ κ^{−1} ⊗ κ^{−1}) ∘ δ(b^{∗}) · (a ⊗ 1))
= (ξ ⊗ ν)((id ⊗ κ) ∘ δ(κ^{−1}(b^{∗})) · (1 ⊗ a))
= (ξ ⊗ ν)((1 ⊗ κ^{−1}(a)) · δ(κ^{−1}(b^{∗})))
= (ξ ⊗ ν)((κ ⊗ id) ∘ δ(κ^{−1}(a)) · (1 ⊗ κ^{−1}(b^{∗})))
= (ν ⊗ ξ)((κ^{−1} ⊗ id) ∘ δ(a) · (κ^{−1}(b^{∗}) ⊗ 1))
= (ν ⊗ ξ)((b^{∗} ⊗ 1) · δ(a)).

By comparing the two identities, we see that R_{ξ^{†}} = R_{ξ}^{†}.

□

By using Lemma 4.3 and applying Theorem 2.1 from [2], we immediately get the following theorem.

Theorem 4.1 Let A be a commutative compact quantum hypergroup. Let P denote the spectrum of the commutative C^{∗}-algebra. Then P is the basis of a normal hypercomplex system L_{1}(P, ν) with a basis unit ϵ.

5 Corepresentations of compact quantum hypergroups and a Peter-Weyl theorem

Let A be a Banach space, and A = (A, δ, ϵ) be a coalgebra [1]. Let V be a Banach space and ι : V → A ⊗ V be a continuous linear map such that

\begin{matrix}(δ ⊗ id) ∘ ι = (id ⊗ ι) ∘ δ, \\ (ϵ ⊗ id) ∘ δ = id, \end{matrix}  (54)

where A ⊗ V denotes the Banach space obtained by completion of the algebraic tensor product with respect to one of the cross-norms [13]. The Banach space V will be called a left comodule over the coalgebra A and (V, ι) — a corepresentation of the coalgebra. If V is finite dimensional, then the corepresentation (V, ι) is called finite dimensional. If (V, ι) is a finite dimensional corepresentation of a coalgebra A and E = {e_{i}|i = 1, …, d} is a basis in V, d = dim V, then ι(e_{i}) = ∑_{j=1}^{d} t_{ij} ⊗ e_{j} for some elements t_{ij} ∈ A which are called matrix elements of the corepresentation (V, ι) with respect to the basis E. For matrix elements t_{ij}, we have the following identities:

\begin{matrix}δ(t_{ij}) = ∑_{k} t_{ik} ⊗ t_{kj}, \\ ϵ(t_{ij}) = δ_{ij}, \end{matrix}  (55)

where δ_{ij} denotes the Kronecker symbol.

Two finite dimensional corepresentations of a coalgebra A, (V_{1}, ι_{1}) and (V_{2}, ι_{2}) are called equivalent if there is an invertible operator F : V_{1} → V_{2} such that

ι_{2} ∘ F = (id ⊗ F) ∘ ι_{1}.  (56)

Let (V_{1}, ι_{1}) and (V_{2}, ι_{2}) be two equivalent finite dimensional corepresentations of a coalgebra A with matrix elements (t^{1}) = (t_{ij}^{1})_{i,j=1}^{d_{1}}, (t^{2}) = (t_{ij}^{2})_{i,j=1}^{d_{2}} with respect to bases in V_{1} and V_{2}. It follows from elementary linear algebra and definition (56) that d^{1} = d^{2} and

(t^{1}) = (f_{ij})′ · (t^{2}) · (f_{ij})′^{−1},  (57)

where (f_{ij}) denotes the matrix of F with respect to the chosen bases in V_{1} and V_{2}, ′ denotes the matrix transpose, and · is the usual matrix multiplication.

A corepresentation (V, ι) of a coalgebra A is called irreducible if there is no proper linear closed subspace V′ ⊂ V such that (V′, ι) is a corepresentation of A. A finite dimensional corepresentation (V, ι) is irreducible if and only if an operator F : V → V such that (id⊗F)∘ι = ι∘F is necessarily a multiple of the identity operator [18].

Definition 5.1 Let A = (A, ·, 1, ∗, δ, ϵ, ⋆, σ_{t}) be a compact quantum hypergroup. We call (V, ι) a corepresentation of A if (V, ι) is a corepresentation of the coalgebra (A, δ, ϵ). All notions concerning corepresentations of a compact quantum hypergroups are understood in the sense of the corresponding notions for its coalgebra structure.

Let (V, ι) be a corepresentation of a compact quantum hypergroup A and let A^{∘} denote the set of all continuous linear functionals on the C^{∗}-algebra A. Then the first formula in (12) defines the structure of an algebra on A^{∘} and the corepresentation (V, ι) gives rise to a representation (V, ι^{∘}) of the algebra A^{∘}, A^{∘} ∋ ξ ↦ ι^{∘}(ξ), defined by the formula

ι^{∘}(ξ) = (ξ ⊗ id) ∘ ι.  (58)

A corepresentation of a compact quantum hypergroup A, (V, ι), is irreducible if and only if such is the representation (V, ι^{∘}) of the algebra A^{∘}. Two finite dimensional corepresentations of A, (V_{1}, ι_{1}) and (V_{2}, ι_{2}), are equivalent if and only if the corresponding representations of A^{∘}, (V_{1}, ι_{1}^{∘}) and (V_{2}, ι_{2}^{∘}) are equivalent.

By using the proof of Lemma 4.8 in [18], we get a similar result.

Lemma 5.1 Let Q_{0} be a finite set of irreducible finite dimensional corepresentations (V^{q}, ι^{q}) of a compact quantum hypergroup A and let λ_{ij}^{q}, q ∈ Q_{0}, i, j = 1, …, d^{q} (d^{q} is the dimension of V^{q}) be an arbitrary set of complex numbers. If t_{ij}^{q} are matrix elements with respect to some bases in V^{q}, then there exists a continuous linear functional α ∈ A^{∘} such that α(t_{ij}^{q}) = λ_{ij}^{q}. Hence the matrix elements are linearly independent.

Proposition 5.1 Let A be a compact quantum hypergroup and (V, ι) be a finite dimensional corepresentation. Then, any element that belongs to the linear span of the matrix elements t_{ij} in any basis of V, is analytic relatively to the one-parameter group σ_{t}.

Proof. Consider the algebra ι^{∘}(A^{∘}). This is a finite dimensional subalgebra of M_{d}(C), the algebra of d × d-matrices over C, d = dim(V). Because, for t ∈ R, σ_{t} is a C^{∗}-algebra automorphism, it is continuous and, hence, for ξ ∈ A^{∘}, we have that σ_{t}^{∘}(ξ) = ξ ∘ σ_{t} ∈ A^{∘}. Identity (39) implies that σ_{t}^{∘} is an automorphism of the algebra A^{∘} for each t ∈ R. This means that ι^{∘} ∘ σ_{t}^{∘} : ι^{∘}(A^{∘}) → ι^{∘}(A^{∘}) is an automorphism. If A_{ι} will denote the Hilbert space of matrix elements of the corepresentation (V, ι) endowed with the inner product defined by (45), then σ_{t} : A_{ι} → A_{ι} and it is easy to check that σ_{t} is a one-parameter group of unitary operators, and so σ_{t} = e^{iNt}, where N : A_{ι} → A_{ι} is a self-adjoint operator, i = \sqrt{−1}, t ∈ ℝ. Let us define σ_{z} : A_{ι} → A_{ι} by σ_{z} = e^{iNz}, z ∈ ℂ. Clearly this is an analytic extension of σ_{t}.

□

Let (V, ι) be a finite dimensional corepresentation of a compact quantum hypergroup A and let A_{ι} be the linear span of the matrix elements. Then A_{ι} is a coalgebra and the restriction mapping ρ_{ι} : A^{∘} → A_{ι}^{∘} defined by ξ ↦ ξ_{ι} = ρ_{ι}(ξ) = ξ|_{A_{ι}} is an epimorphism of the algebra A^{∘} onto the algebra A_{ι}^{∘} and ι^{∘} is a representation of A_{ι}^{∘} on the linear space V.

Let now H be a finite dimensional Hilbert space with an inner product (·, ·) and let (H, ι) be a corepresentation of a compact quantum hypergroup A. Since, by Proposition 5.1, matrix elements of a finite dimensional corepresentation are analytic, we can make the following definition.

Definition 5.2 A finite dimensional corepresentation (H, ι) of a compact quantum hypergroup A is called a ^{†}-corepresentation if, for all u, v ∈ H, we have

∑_{i=1}^{d}(u, v_{i})b_{i} = ∑_{i=1}^{d}(u_{i}, v)a_{i}^{†},  (59)

where ι(u) = ∑_{i=1}^{d} a_{i} ⊗ u_{i}, ι(v) = ∑_{i=1}^{d} b_{i} ⊗ v_{i}, a_{i}, b_{i} ∈ A, u_{i}, v_{i} ∈ H, and d = dim H.

Lemma 5.2 Let t_{ij}, i, j = 1, …, d, be matrix elements of a finite dimensional ^{†}-corepresentation (H, ι) with respect to an orthonormal in H. Then

t_{ij}^{†} = t_{ji}.  (60)

Proof. The claim immediately follows from (59) by setting u = e_{i}, v = e_{j}. □

Lemma 5.3 Let (H, ι) be a finite dimensional corepresentation of a compact quantum hypergroup A. Define an involution ξ_{ι} ↦ ξ_{ι}^{†} on A_{ι}^{∘} by

ξ_{ι}^{†}(a) = \overline{ξ}_{ι}(a^{†}),  a ∈ A_{ι}.  (61)

Then ξ_{ι}^{†} is a continuous functional on A_{ι} and the corepresentation (H, ι) is a ^{†}-corepresentation if and only if ι^{∘} is a ^{†}-representation of the involutive algebra A_{ι}^{∘}.

Proof. Because A_{ι} is finite dimensional and ξ_{ι}^{†} is a linear mapping, it is continuous. Now, for ξ_{ι} ∈ A_{ι}^{∘}, u, v ∈ A_{ι}^{∘} with ι(u) = ∑_{i=1}^{d} a_{i} ⊗ u_{i}, ι(v) = ∑_{i=1}^{d} b_{i} ⊗ v_{i}, we have

(ι^{∘}(ξ_{ι})(u), v) = (∑_{i=1}^{d}(ξ_{ι}(a_{i})u_{i}, v) = ξ (∑_{i=1}^{d}(u_{i}, v)a_{i}) .

On the other hand,

(u, ι^{∘}(ξ_{ι}^{†})(v)) = (u, ∑_{i=1}^{d} ξ_{ι}^{†}(b_{i})v_{i}) = ∑_{i=1}^{d}(u, v_{i})\overline{ξ_{ι}^{†}}(b_{i}) = ξ_{ι} (∑_{i=1}^{d}(u, v_{i})b_{i}^{†}) .

The proof follows if we compare these two identities.

□

Lemma 5.4 Let (H, ι) be a finite dimensional ^{†}-corepresentation of a compact quantum hypergroup A. Then (H, ι) is a finite direct sum of irreducible finite dimensional ^{†}-corepresentations, i.e. H = ⊕_{i=1}^{k}H_{i} and (H_{i}, ι_{i}) is an irreducible ^{†}-corepresentation with ι_{i} = ι|_{H_{i}}.

Proof. Clearly it will be sufficient to show that, for an invariant subspace H_{1}, the subspace H_{1}^{⊥} = H ⊖ H_{1} will be invariant, i.e. ι : H_{1}^{⊥} → A ⊗ H_{1}^{⊥}. Choose an orthonormal basis in H_{1}, e_{1}, …, e_{m}, and let e_{m+1}, …, e_{m+n} be an orthonormal basis in H_{1}^{⊥}. Set u = e_{i}, v = e_{m+j}, where i ≤ m, j ≤ n. Then we have that

ι(u) = ι(e_{i}) = ∑_{k=1}^{m} t_{ik} ⊗ e_{k}

since H_{1} is an invariant subspace, and

ι(v) = ι(e_{m+j}) = ∑_{k=1}^{m+n} t_{m+j k} ⊗ e_{k}.

By using (59), we find that

t_{m+j i} = ∑_{k=1}^{m+n}(e_{i}, e_{k})t_{m+j k} = ∑_{k=1}^{m}(e_{k}, e_{m+j})t_{ik}^{†} = 0

for all i = 1, … m, j = 1, …, n. This shows that H_{1}^{⊥} is invariant.

□

We now prove some orthogonality relations.

Theorem 5.1 Let (V^{p}, ι^{p}) and (V^{q}, ι^{q}) be finite dimensional irreducible corepresentations of a compact quantum hypergroup A. Let t_{ij}^{p} and t_{kl}^{q} denote matrix elements of the corresponding corepresentations. Then

ν(t_{ij}^{p}κ(t_{kl}^{q})) = 0  (62)

if either the corepresentations are not equivalent or i ≠ l.

Proof. Let us apply the strong invariance condition for ν given by (43) to elements a = t_{ij}^{p} and b = κ(t_{kl}^{q}). For these elements, the left-hand side of (43) becomes:

(id ⊗ ν)((κ ⊗ id) ∘ δ(t_{ij}^{p}) · (1 ⊗ κ(t_{kl}^{q}))) = ∑_{r} ν(t_{rj}^{p}κ(t_{kl}^{q}))κ(t_{ir}^{p}),

whereas the right-hand side will be

(id ⊗ ν)((1 ⊗ t_{ij}^{p}) · δ(κ(t_{kl}^{q}))) = (id ⊗ ν)((1 ⊗ t_{ij}^{p}) · Π(∑_{s} κ(t_{ks}^{q}) ⊗ κ(t_{sl}^{q})))
= (id ⊗ ν)((1 ⊗ t_{ij}^{p}) · ∑_{s} κ(t_{sl}^{q}) ⊗ κ(t_{ks}^{q}))
= ∑_{s} ν(t_{ij}^{p}κ(t_{ks}^{q}))κ(t_{sl}^{q}).

Now, if we recall that the matrix elements of corepresentations are linearly independent and κ is invertible, comparing the last two expressions we see that relation (62) holds.

□

Remark 5.1 If the corepresentations (V^{p}, ι^{p}) and (V^{q}, ι^{q}) are ^{†}-corepresen-tations then, by using (60), we can rewrite (62) in the following form

ν(t_{ij}^{p}t_{lk}^{q}^{∗}) = 0.  (63)

The following proposition is a modification of the well-known Hilbert-Schmidt theorem.

Proposition 5.2 Let the operator T_{a}, a ∈ A_{0}, a^{†} = a, be defined by (46). Let y = T_{a}(x) for some x ∈ H_{ν} and

y = ∑_{i=1}^{∞}⟨y, v^{λ_{i}}⟩v^{λ_{i}}  (64)

be the Fourier expansion of y with respect to an orthonormal set of the eigen vectors v^{λ_{i}} of the self-adjoint compact operator T_{a}, where λ_{i} is a corresponding eigen value, λ_{i} ≠ 0. Then v^{λ} ∈ A and the series (64) converges the C^{∗}-norm.

Proof. First of all, if v^{λ_{i}} is an eigen vector corresponding to an eigen value λ_{i} ≠ 0, then by Proposition 4.1 (b), v^{λ} = \frac{1}{λ}T_{a}(v^{λ}) ∈ Ran T_{a} ⊂ A. Hence v^{λ_{i}} ∈ A.

Now, to prove convergence of (64), we use the Cauchy criterion. So let m, n ∈ Z_{+}. We have

∥ ∑_{i=m}^{m+n}⟨y, v^{λ_{i}}⟩v^{λ_{i}}∥ = ∥ ∑_{i=m}^{m+n}⟨T_{a}(x), v^{λ_{i}}⟩v^{λ_{i}}∥ = ∥ ∑_{i=m}^{m+n}⟨x, T_{a}(v^{λ_{i}})⟩v^{λ_{i}}∥
= ∥ ∑_{i=m}^{m+n}⟨x, v^{λ_{i}}⟩λ_{i}v^{λ_{i}}∥ = ∥ ∑_{i=m}^{m+n}⟨x, v^{λ_{i}}⟩T_{a}(v^{λ_{i}})∥
= ∥T_{a}( ∑_{i=m}^{m+n}⟨x, v^{λ_{i}}⟩v^{λ_{i}})∥ ≤ ∥a∥ ∥ ∑_{i=m}^{m+n}⟨x, v^{λ_{i}}⟩v^{λ_{i}}∥_{ν}.

In order to obtain the last inequality, we used estimate (47). Finally, since x ∈ H_{ν}, ∥ ∑_{i=m}^{m+n}⟨x, v^{λ_{i}}⟩v^{λ_{i}}∥_{ν} → 0 as m → ∞ and hence series (64) converges in the C^{∗}-norm. It clearly converges to y.

□

Theorem 5.2 Let Q be the set of all finite dimensional irreducible nonequivalent ^{†}-corepresentations (V^{q}, ι^{q}), q ∈ Q, of a compact quantum hypergroup A and B = {t_{ij}^{q} : q ∈ Q, i, j = 1, …, d_{q} = dim V^{q}} be the set of all matrix elements of these corepresentations with respect to some bases. Then the linear span of the set B is dense in A with respect to the C^{∗}-norm.

Proof. Let us again consider the operator T_{a} : H_{ν} → H_{ν}. Because this operator is compact and self-adjoint,

Ran T_{a} = ⨁_{λ≠0} V^{λ},  (65)

where V^{λ} is the finite dimensional eigen space corresponding to the eigen value λ.

Let us show that

δ(V^{λ}) ⊂ A ⊗ V^{λ}.  (66)

It follows from Proposition 5.2 that V^{λ} ⊂ A and hence δ : V^{λ} → A⊗A. Now, to prove (66), let v^{λ} ∈ V^{λ} and let φ be an arbitrary continuous functional on A. It will suffice to prove that u = (φ ⊗ id) ∘ δ(v^{λ}) ∈ V^{λ}. To see that, by using coassociativity of δ, we find

T_{a}(u) = (id ⊗ ν((1 ⊗ a) · δ(u))
= (id ⊗ ν)((1 ⊗ a) · δ((φ ⊗ id) ∘ δ(v^{λ})))
= (id ⊗ ν)((1 ⊗ a) · (φ ⊗ δ) ∘ δ(v^{λ}))
= (φ ⊗ id ⊗ ν)((1 ⊗ 1 ⊗ a) · (id ⊗ δ) ∘ δ(v^{λ}))
= (φ ⊗ id ⊗ ν)((1 ⊗ 1 ⊗ a) · (δ ⊗ id) ∘ δ(v^{λ}))
= (φ ⊗ id) ∘ δ((id ⊗ ν)((1 ⊗ a) · δ(v^{λ})))
= (φ ⊗ id) ∘ δ(T_{a}(v^{λ})) = λ u.

This proves (66), which means that V^{λ} = (V^{λ}, δ|_{V^{λ}}) is a finite dimensional corepresentation.

Let us now show that (V^{λ}, δ|_{V^{λ}}) is a ^{†}-corepresentation. Equip V^{λ} with the inner product defined by

(u, v) = ν(uv^{∗}).

Then condition (59) can be written as

(id ⊗ ν)((1 ⊗ u) · (id ⊗ ∗) ∘ δ(v)) = (id ⊗ ν)((† ⊗ id) ∘ δ(u) · (1 ⊗ v^{∗}))

By applying the map (∗ ⊗ id) to both sides of this equality, we obtain

(id ⊗ ν)((1 ⊗ u) · δ(v^{∗})) = (id ⊗ ν)((κ ⊗ id) ∘ δ(u) · (1 ⊗ v^{∗})),

which is just the strong invariance condition (43).

Next, if we apply the map (id ⊗ ϵ) to the right-hand side of (66), we see that V^{λ} is contained in the linear span of the matrix elements of the corepresentation V^{λ}, which because of (65) and Proposition 5.2, implies that the linear span of matrix elements of V^{λ} for all eigen values λ ≠ 0 is dense in Ran T_{a}. So now application of Proposition 4.2 yields the claim.

□

The following corollary is immediate.

Corollary 5.1 Let B be defined as in Theorem 5.2. Then the linear span of B is total in H_{ν} with respect to the L_{2}-norm.

Theorem 5.3 Let V be a Banach space, (V, ι) be an irreducible corepresentation of a compact quantum hypergroup A. Then V is finite dimensional.

Proof. Consider the finite dimensional linear spaces A_{q} formed by the linear span of matrix elements of the finite dimensional ^{†}-corepresentations (V^{q}, ι^{q}), and let P_{q} : A → A_{q} denotes the orthogonal projection in H_{ν} onto the subspace A_{q}. Then clearly P_{q} is continuous in the C^{∗}-norm and

δ ∘ P_{q} = (P_{q} ⊗ id) ∘ δ = (id ⊗ P_{q}) ∘ δ = (P_{q} ⊗ P_{q}) ∘ δ.  (67)

Choose an arbitrary vector v ∈ V and a finite dimensional corepresentation (V^{q}, ι^{q}) such that (P_{q} ⊗ id) ∘ ι(v) ≠ {0}. This is always possible since, for an arbitrary v ∈ V and v ≠ 0, ι(v) ≠ 0, and hence, if ι(v) = ∑ a_{i} ⊗ v_{i} with a_{i} ∈ A and linearly independent v_{i} ∈ V, there is an index i_{0} such that a_{i_{0}} ≠ 0. By using Corollary 5.1, we can expand a_{i_{0}} in H_{ν} with respect to matrix elements t_{ij}^{q} ∈ A_{ι^{q}}, and thus find a projection P_{q} with P_{q}(a_{i_{0}}) ≠ 0.

Now, let V_{v} = {(ξ ∘ P_{q} ⊗ id) ∘ ι(v)|ξ ∈ A^{∘}}, where A^{∘} denotes the linear space of all continuous functionals on A. It is clear that V_{v} is finite dimensional since the linear space {ξ ∘ P_{q}|ξ ∈ S^{∘}} is such. We will prove that V_{v} is an invariant subspace of the corepresentation (V, ι), i.e. ι(V_{v}) ⊆ A ⊗ V_{v}.

Let u_{ξ} = (ξ ∘ P_{q} ⊗ id) ∘ ι(v) and φ ∈ A^{∘} be a continuous functional on the C^{∗}-algebra A. Then

(φ ⊗ id) ∘ ι(u_{ξ}) = (φ ⊗ id) ∘ ι ∘ (ξ ∘ P_{q} ⊗ id) ∘ ι(v)
= (φ ⊗ ξ ⊗ id) ∘ (id ⊗ P_{q} ⊗ id) ∘ (id ⊗ ι) ∘ ι(v)
= (φ ⊗ ξ ⊗ id) ∘ (id ⊗ P_{q} ⊗ id) ∘ (δ ⊗ id) ∘ ι(v)
= (φ ⊗ ξ ⊗ id) ∘ (δ ⊗ id) ∘ (P_{q} ⊗ id) ∘ ι(v)
= (φ · ξ ∘ P_{q} ⊗ id) ∘ ι(v) = u_{φ·ξ}

is in V_{v} and, hence, it is invariant. Clearly, it is closed and, because the corepresentation was assumed to be irreducible, V_{v} = V.

□

References

[1] E. Abe. Hopf algebras, volume 74 of Cambridge Tracts in Math. Cambridge University Press, 1980.

[2] Yu. M. Berezanskii and A. A. Kaluzhnii. Harmonic analysis in hypercomplex systems. Naukova Dumka, Kiev, 1992. Russian (To be translated into English).

[3] K. Bragiel. On the spherical and zonal spherical functions on a compact quantum group. Letters in Math. Phys., 22:195–201, 1991.

[4] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1. Springer-Verlag, New York, Heidelberg, Berlin, 1979.

[5] Y. A. Chapovsky and L. I. Vainerman. Hypergroup structures associated with a pair of quantum groups (SU_{q}(n), U_{q}(n − 1)). In Methods of Functional Analysis in Problems of Mathematical Physics, pages 47–69, 1992.

[6] Yu. A. Chapovsky and L. I. Vainerman. A Gelfand pair of compact quantum groups. Funksional’nyi Analiz i Ego Prilozheniya, 29(2):67– 71, 1995.

[7] M. Enock and J.-M. Schwartz. Kac algebras and duality of locally compact groups. Springer-Verlag, New York, Heidelberg, Berlin, 1992.

[8] J. Faraut. Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques. In Analyse Harmonique, pages 315–446. CIMPA, Nice, 1982.

[9] R. I. Jewett. Spaces with an abstract convolution of measures. Adv. Math., 18:1–101, 1975.

[10] S. V. Kerov. Duality of finite dimensional ∗-algebras. Vestnik Leningradskogo Universiteta, 7:23–29, 1974. Russian.

[11] T. H. Koornwinder. Discrete hypergroups associated with compact quantum Gelfand pairs. Contemporary Mathematics, 183:213–235, 1995.

[12] T. Masuda and Y. Nakagami. A von Neumann algebra framework for duality of the quantum groups. Publ. RIMS, 30(5):799–850, 1994.

[13] M. Takesaki. Theory of Operator Algebras. New York, Heidelberg, Berlin. Springer-Verlag, 1979.

[14] L. I. Vainerman. Gelfand pairs of quantum groups, hypergroups and q-special functions. Contemporary Mathematics, 183:373–394, 1995.

[15] L. I. Vainerman. Hypergroup structures associated with Gel’fand pairs of compact quantum groups. Asterisque, 232:231–242, 1995.

[16] L. I. Vainerman and G. I. Kac. Nonunimodular ring groups and Hopfvon Neumann algebras. Mat. Sb., 94(136):194–225, 1974. English transl. in Math. USSR-Sb. 23:185–214, 1974.

[17] L. I. Vainerman and G. B. Podkolzin. Quantum Shtieffel manifold and double cosets of quantum unitary group. Preprint 96-14, Université d’Orléans, 1996 (To appear in Pacific Journal of Math.).

[18] S. L. Woronowicz. Compact matrix pseudogroups. Commun. Math. Phys., 111:613–665, 1987.

[19] S. L. Woronowicz. Compact quantum groups. Preprint, 1993.