On Quantum Duality of Group Amenability

: In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.


Introduction
Given a compact group G, denoted by C(G) the C*-algebra of continuous functions on G, one can define a morphism ∆ : C(G) → C(G) ⊗ C(G), by ∆( f )(g 1 , g 2 ) = f (g 1 g 2 ), where f ∈ C(G), g 1 , g 2 ∈ G, and C(G) ⊗ C(G) is naturally identified with C(G × G), which satisfies the co-associativity The morphism ∆ is called a co-multiplication on C(G), under which the pair (C(G), ∆) comes into being a compact quantum group defined in the sense of Woronowicz [1].
Then, the pair (A, ∆) is called a compact quantum group (CQG).
For an arbitrary CQG (A, ∆), by [2], there exists a unique state h A on A so that for all a ∈ A, (id ⊗ h A )∆(a) = (h A ⊗ id)∆(a) = h A (a)1, which is called the Haar integral of (A, ∆). For the commutative CQG (C(G), ∆) associated to a classical compact group G described as above, the Haar integral h C(G) is the integral with respect to the Haar measure on G, which has full support and, therefore, is faithful. However, the Haar integral on an arbitrary CQG (A, ∆) needs not be always faithful. Each CQG (A, ∆) has a canonical dense Hopf *-subalgebra (A 0 , ∆ 0 ) linearly spanned by matrix entries of all finite dimensional co-representations of (A, ∆), where ∆ 0 is given by restricting the co-multiplication ∆ from A to A 0 . In the article, we call (A 0 , ∆ 0 ) the associated algebraic CQG of (A, ∆) (algCQG).
Let Γ be a discrete group, and let C * r (Γ) and C * (Γ) be its reduced and full group C*-algebras. Γ is called amenable if there exists an invariant mean on L ∞ (Γ). Endowed with co-multiplications ∆ r and ∆, (C * r (Γ), ∆ r ) and (C * (Γ), ∆) come into being CQGs, which are called reduced and universal CQG, respectively. The Haar integral of (C * r (Γ), ∆ r ) is faithful, but that of (C * (Γ), ∆) may not be; the co-unit of (C * (Γ), ∆) is norm-bounded, but that of (C * r (Γ), ∆ r ) may not be. From [3], the Haar integral of (C * (Γ), ∆) is faithful if and only if the co-unit of (C * (Γ), ∆) is norm-bounded if and only if Γ is amenable. Under what conditions is the Haar integral on a CQG faithful and the co-unit norm-bounded? In [3], Bédos, Murphy, and Tuset defined the co-amenability of CQG, which can induce the faithfulness of its Haar integral and the norm-boundness of its co-unit. As the quantum dual of group amenability, (C * r (Γ), ∆ r ) is co-amenable if and only if Γ is amenable. Denote C[Γ] the group algebra of Γ equipped with its canonical Hopf *-algebra structure. By [3], C * r (Γ) and C * (Γ) are the CQG completions of C [Γ]. Under what conditions, for an arbitrary algCQG (A 0 , ∆ 0 ), is the CQG completion of (A 0 , ∆ 0 ) unique? Generally, it is not unique. However, in the co-amenable case, the answer is affirmative [3]. Moreover, in [4,5], Bédos, Murphy, and Tuset studied the amenability and co-amenability of algebraic quantum groups, a sufficient large quantum group class including CQGs and discrete quantum groups(DQGs), which admits a dual that is also an algebraic quantum group.
In the group case, a product of two discrete amenable groups is amenable; as a quantum counterpart, co-amenability is preserved under formulation of the tensor product of two CQGs [3]. In [6], we constructed the reduced and universal quantum double of two dually paired CQGs. Since the tensor product of two CQGs is a special case of quantum double of CQGs when the pairing is trivial, inspired by the underlying stability of co-amenability of CQGs and the symmetrical idea, in the article, we will focus on studying the stability of the co-amenability in the process of quantum double constructions. In Section 2, we first recall the definition of co-amenability of compact quantum groups, as well as some related properties, and then briefly present the quantum double construction procedure. By symmetric calculations, as used in the case of the group amenability, in Section 3, we show that the quantum double of CQGs is unique when the paired CQGs are both co-amenable and that co-amenability is preserved under formulation of the quantum double constructions of CQGs. Using this result, one can yield a co-amenable new CQG from a pair of co-amenable CQGs.

Preliminaries
In this section, we first recall the definition of co-amenability of CQGs and some of its properties. Let (A, ∆) be a CQG, (A 0 , ∆ 0 ) be the associated algCQG of (A, ∆), and h the Haar integral of (A, ∆). As is well known, h is faithful on (A 0 , ∆ 0 ) but need not be faithful on the C*-algebra (A, ∆). Set where N h is the left kernel of h. Then, A r becomes a CQG, where its co-multiplication ∆ r is defined as for all a ∈ A, where η : A → A r is the canonical map. (A r , ∆ r ) is called the reduced quantum group of (A, ∆), where its co-unit ε r , antipode S r , and Haar state h r are determined by respectively. What needs to be pointed out is that the co-unit ε r of (A r , ∆ r ) is faithful. However, generally, the co-unit ε r needs not be norm-bounded.
With the following proposition, one can obtain the co-amenability of (A, ∆) without reference to the reduced quantum group (A r , ∆ r ).

Proposition 1 ([3]
). Let (A, ∆) be a CQG, and h and ε be its Haar integral and co-unit, respectively. Then, (A, ∆) is co-amenable if and only h is faithful and ε is norm-bounded.
Assume that (A, ∆) and (A 0 , ∆ 0 ) are described as above. Let · c be a C*-norm on (A 0 , ∆ 0 ), and let (A c , ∆ c ) be a compact quantum group completion of (A 0 , where the variable π travels over all unital *-representations π of A 0 . It is not difficult to find that · u is the greatest regular C*-norm on A 0 . Denote A u as the C*-algebra completion of A 0 with respect to · u and ∆ u the extension to A u of ∆. Then, (A u , ∆ u ) is a CQG, which is called the universal quantum group of (A, ∆). Define · r on A 0 as a r := η(a) , for all a ∈ A 0 , which is the least regular C*-norm on A 0 . Then, the underlying A r is the C*-algebraic completion of A 0 with respect to · r .
Let (A, ∆ A ) and (B, ∆ B ) be two dually paired CQGs, and let (A 0 , ∆ A 0 ) and (B 0 , ∆ B 0 ) be described as above. Denote by A 0 B 0 . It is well known that A 0 B 0 , the algebraic tensor product of A 0 and B 0 , can be made into a linear space in a natural way. Under the multiplication map, m D and involution * D on A 0 B 0 defined as the following: turn into a non-degenerate associative * −algebra, which is similar to the classical Drinfeld's quantum double [18] in the pure algebra level, and then we denote it by D(A 0 , B 0 ). To avoid using too many brackets, we will simplify m D ((a, b)(a , b )) as (a, b)(a , b ) and simplify S(a) as Sa in sequel. Under the structure maps,     Proof. Suppose that (A 0 , ∆ A 0 ) and (B 0 , ∆ B 0 ) are the associated algCQGs, respectively. Let · c be a regular C*-norm and A c be the CQG completion of (A 0 , ∆ A 0 ). As described in Section 2, A u and A r are both CQG completions of (A 0 , ∆ A 0 ). Because A is co-amenable, by Proposition 2 (ii), there is a unique CQG completion for the associated algCQGs (A 0 , ∆ A 0 ). Hence, for all a ∈ A 0 and b ∈ B 0 . Combining with the equations A r = A u and B r = B u , one can symmetrically obtain that a r = a u , b r = b u .
on A 0 and B 0 . Moreover, Equation (1) also holds on A 0 B 0 . In fact, for any C*-norm · on A 0 B 0 , we have for all a ∈ A 0 , b ∈ B 0 . Then, Considering the multiplication rule on the quantum double D(A 0 , B 0 ) ([6]), for any (a, b) ∈ D(A 0 , B 0 ), From the above expression Equation (2), one can find that each element (a, b) in D(A 0 , B 0 ) is a linear combination of elements as c ⊗ d ∈ A 0 B 0 . By the discussion in the underlying paragraph, we have where a (2) and b (2) are as presented in Equation (2), which induces that In sequel, (D u (A, B), ∆ D u ) and (D r (A, B), ∆ D r ) will be denoted by (D(A, B), ∆ D ). (D(A, B), ∆ D ) be the quantum double of (A, ∆ A ) and (B, ∆ B ) based on a non-degenerate compact quantum group pairing (A, B, ·, · ). Assume that (A, ∆ A ) and (B, ∆ B ) are both co-amenable. Then, (D(A, B), ∆ D ) is co-amenable.

Theorem 2. Let
Proof. By Proposition 1, we have to prove that the following two conditions hold.
Denote bb * by k; then, we can obtain that Again, for all (c, d) ∈ D(A 0 , B 0 ), one can get and where (a, b) ∈ D(A 0 , B 0 ), α s are in some index set, and the limit is taken with respect to the universal C*-norm · u on D(A 0 , B 0 ). Thus, (a , b ) can be rewritten as the following: where ) = 0.
Because (A, ∆ A ) and (B, ∆ B ) are both co-amenable, by Proposition 1, h A and h B are both faithful. Hence, h A u and h B u are also faithful. Combining with the underlying equation, we obtain that a (2) = 0 and b (2) = 0; thus, by (5), we get (a , b ) = 0, which states that h D u is faithful on D u (A, B).
(ii) The co-unit of D (A, B) is norm-bounded. First, we show that ε D 0 defined as before Proposition 3 is a *-homomorphism. Using the definition of ε D 0 , we have Let ε A and ε B be the co-units on A and B, respectively. For all (a, b) ∈ D(A, B), we define which can be regarded as the extension to (D(A, B), ∆ D ) of ε D 0 .
Considering the continuity of extension of ε D 0 from D(A 0 , B 0 ) to D (A, B), ε D is a *-homomorphism and then the co-unit on D (A, B).
To prove that the co-unit ε D on D(A, B) is norm-bounded, it suffices to show that the Haar integral ε D r of D r (A, B) is norm-bounded with respect to the supremum norm, since the co-unit of where a (2) is in A r \ A 0 or b (2) is in B r \ B 0 . Since A and B are co-amenable, by Proposition 1, ε A and ε B are both norm-bounded. Hence, ε A r and ε B r are norm-bounded, i.e., there exist two positive number M A and M B such that Thus,