L p -improving convolution operators on finite quantum groups
aa r X i v : . [ m a t h . OA ] M a y L p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUMGROUPS SIMENG WANG
Abstract.
We characterize positive convolution operators on a finite quantum group G whichare L p -improving. More precisely, we prove that the convolution operator T ϕ : x ϕ ⋆ x givenby a state ϕ on C ( G ) satisfies ∃ < p < , k T ϕ : L p ( G ) → L ( G ) k = 1if and only if the Fourier series ˆ ϕ satisfy k ˆ ϕ ( α ) k < α , and if and only if the state ( ϕ ◦ S ) ⋆ ϕ is non-degenerate (where S is the antipode).We also prove that these L p -improving properties are stable under taking free products, whichgives a method to construct L p -improving multipliers on infinite compact quantum groups. Ourmethods for non-degenerate states yield a general formula for computing idempotent statesassociated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski. Introduction
The convolution operators or multipliers constitute a central part of Fourier analysis. Oneamong phenomena studied on the circle group T is the existence and behavior of positive Borelmeasures that convolve L p ( T ) into L q ( T ) with finite q > p for a given 1 < p < ∞ , which areconsidered to be L p -improving measures. An example due to Oberlin [Obe82] is the Cantor-Lebesgue measure supported by the usual middle-third Cantor set. Oberlin revealed that, after acareful analysis on the structure of this measure, this result can be reduced to proving that thereexists p < k µ ⋆ f k ≤ k f k p , f ∈ L p ( Z / Z )where the L p -norms are those taken with respect to the normalized counting measure on the cyclicgroup Z / Z = { , , } with three elements and µ is the probability measure with mass 1 / G is an arbitrary finitegroup and T µ : f µ ⋆ f is the convolution operator associated to a probability measure µ on G ,then ( ∃ p < , k T µ : L p ( G ) → L ( G ) k = 1) ⇔ G = h ij − : i, j ∈ supp µ i , which provides a more general method to construct L p -improving measures on groups.In this paper we give an alternative approach related to these topics, in the context of quantumgroups and noncommutative L p -spaces. We show that, for a finite quantum group G and a state ϕ on C ( G ), denoting by ˆ ϕ the Fourier series of ϕ and writing ψ = ( ϕ ◦ S ) ⋆ ϕ , S being the antipode,the following assertions are equivalent (Theorem 4.4):(1) there exists 1 < p < ∀ x ∈ C ( G ) , k ϕ ⋆ x k ≤ k x k p ;(2) k ˆ ϕ ( α ) k < α ∈ Irr( G ) \ { } ;(3) For any nonzero x ∈ C ( G ) + , there exists n ≥ ψ ⋆n ( x ) > ϕ “generates” thequantum group G , which will be explained in the last section. We will illustrate by example inRemark 4.7 that the finiteness condition in the above conclusion is rather crucial and cannot beremoved. Mathematics Subject Classification.
Primary: 20G42, 46L89. Secondary: 43A22, 46L30, 46L51.
Key words and phrases. L p -improving operators, compact quantum groups, positive convolution operators. In particular, the result characterizes the Fourier-Schur multipliers on finite groups which havean L p -improving property. Let Γ be a finite group and ϕ be a positive definite function on Γ. Let M ϕ be the associated Fourier-Schur multiplier operator determined by M ϕ ( λ ( γ )) = ϕ ( γ ) λ ( γ ) forall γ ∈ Γ. Then ∃ < p < , k M ϕ x k ≤ k x k p , x ∈ C ∗ (Γ)if and only if | ϕ ( γ ) | < γ ∈ Γ \ { e } .We should emphasize that our argument relies essentially on new and interesting propertieson the unital trace preserving operators on noncommutative L p -spaces, based on the recent workof Ricard and Xu [RX16]. In fact, the following fact proved in Theorem 1.6 plays a key role inour argument. For a finite dimensional C*-algebra A equipped with a faithful tracial state τ , and T : A → A a unital trace preserving map, the L p -improving property(0.1) ∃ < p < , k T : L p ( A ) → L ( A ) k = 1holds if and only if we have the following “spectral gap”:sup x ∈ A \{ } ,τ ( x )=0 k T x k k x k < . We provide two proofs of this result, where one is based on very elementary arguments with anadditional assumption of 2-positivity and another, which is rather short, on [RX16]. In Theorem1.9 we also show that the L p -improving property (0.1) remains stable under the free products.This method permits us to give L p -improving convolution operators for infinite quantum groups.In this paper we also include some simple properties of non-degenerate states on compactquantum groups with applications. We prove in Lemma 3.3 that the convolution Ces`aro limitof a non-degenerate state is the Haar state, which not only contributes to the proof of our mainresult, but also yields a generalization of [BFS12, Theorem 2.2] concerning the computation ofidempotent states associated to Hopf images.We end this introduction with a brief description of the organization of the paper. Section 1deals with the characterization of unital trace preserving L p -improving operators on finite dimen-sional C*-algebras and their free products. In Section 2 we present some preliminaries on compactquantum groups and the related Fourier analysis. Here we give a short and explicit calculation ofFourier series for compact quantum groups, parallel to the case of classical compact groups, whichdoes not exist in other literature. In Section 3 we obtain some properties of non-degenerate stateson a general compact quantum group. The last Section 4 is devoted to the positive convolutionoperators on finite quantum groups, and constructions of operators with similar properties oninfinite compact quantum groups by free product.1. L p -improvement and spectral gaps Basic notions.
Let us firstly present some preliminaries and notations on noncommutative L p -spaces and free products for later use. All the facts mentioned below are well-known.1.1.1. Noncommutative L p -spaces. Here we recall some basics of noncommutative L p -spaces onfinite von Neumann algebras. We refer to [Tak02] for the theory of von Neumann algebras andto [PX03] for more information on noncommutative L p -spaces. Let M be a finite von Neumannalgebra equipped with a normal faithful tracial state τ . Let 1 ≤ p < ∞ . For each x ∈ M , wedefine k x k p = [ τ ( | x | p )] /p . One can show that kk p is a norm on M . The completion of ( M , kk p ) is denoted by L p ( M , τ ) orsimply by L p ( M ). The elements of L p ( M ) can be described by densely defined closed operatorsmeasurable with respect to ( M , τ ), as in the commutative case. For convenience, we set L ∞ ( M ) = M equipped with the operator norm. Since | τ ( x ) | ≤ k x k for all x ∈ M , τ extends to a continuousfunctional on L ( M ). Let 1 ≤ p, q, r ≤ ∞ be such that 1 /p + 1 /q = 1 /r . If x ∈ L p ( M ) and y ∈ L q ( M ), then xy ∈ L r ( M ) and the following H¨older inequality holds: k xy k r ≤ k x k p k y k q . p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 3 In particular, if r = 1, | τ ( xy ) | ≤ k xy k ≤ k x k p k y k q for arbitrary x ∈ L p ( M ) and y ∈ L q ( M ).This defines a natural duality between L p ( M ) and L q ( M ): h x, y i = τ ( xy ). For any 1 ≤ p < ∞ we have L p ( M ) ∗ = L q ( M ) isometrically.1.1.2. Free products.
We firstly recall some constructions of free product of C*-algebras, for whichwe refer to [VDN92] and [NS06] for details. Consider a family of unital C*-algebras ( A i , φ i ) i ∈ I with distinguished faithful states φ i and associated GNS constructions ( π i , H i ). Set ˚ A i = ker φ i and ˚ a i = a i − φ i ( a i )1 for each i and a i ∈ A i . Construct a vector space(1.1) A = C ⊕ M n ≥ (cid:16) M i = i = ···6 = i n ˚ A i ⊗ ˚ A i ⊗ · · · ⊗ ˚ A i n (cid:17) . We equip A with an algebra structure such that is the identity and the multiplication of aletter a ∈ ˚ A i with an elementary tensor a ⊗ a ⊗ · · · ⊗ a n in ˚ A i ⊗ ˚ A i ⊗ · · · ⊗ ˚ A i n is defined as a · ( a ⊗ a ⊗ · · · ⊗ a n ) = a ⊗ a ⊗ a ⊗ · · · ⊗ a n , if i = i ,˚( aa ) ⊗ a ⊗ a ⊗ · · · ⊗ a n + φ i ( aa ) a ⊗ · · · ⊗ a n , if i = i .Moreover, we give an involution on A by( a ⊗ a ⊗ · · · ⊗ a n ) ∗ = a ∗ n ⊗ a ∗ ⊗ · · · ⊗ a ∗ . In this sense A becomes a ∗ -algebra, and each A i can be viewed as a ∗ -subalgebra in A byidentifying A i with C ⊕ ˚ A i in the big direct sum. We call A the algebraic free product of ( A i ) i ∈ I .It then can be shown that the algebra A admits a faithful ∗ -representation ( π, H, ξ ) such that π | A i = π i for each i ∈ I and φ ( · ) := h π ( · ) ξ, ξ i restricted on A i coincides with φ i . Moreover thestate φ is faithful on A . Then the reduced C*-algebraic free product of ( A i ) i ∈ I is the C*-algebragenerated by π ( A ) in B ( H ), i.e., the norm closure of π ( A ) in B ( H ), denoted by ∗ c i ∈ I A i ; and thestate extends to ∗ c i ∈ I A i , called the free product state of ( φ i ) i ∈ I and denoted by ∗ i ∈ I φ i . If moreovereach A i = M i is a von Neumann algebra and each φ i is normal, then the weak closure of π ( A ) in B ( H ), is defined to be the von Neumann algebraic free product of ( M i ) i ∈ I , denoted by ¯ ∗ i ∈ I M i ,and the free product state φ = ∗ i ∈ I φ i is also normal. Also, we remark that if each φ i is a tracialstate, then φ = ∗ i ∈ I φ i is also tracial.Let A i and B i be unital C*-algebras with distinguished faithful states φ i and ψ i ( i ∈ I )respectively, and let T i : A i → B i be a unital state preserving map for each i ∈ I . Set( A, φ ) = ∗ i ∈ I ( A i , φ i ) and ( B, ψ ) = ∗ i ∈ I ( B i , ψ i ). Then it is obvious that T ( a a · · · a n ) = T i ( a ) · · · T i n ( a n ) ( a k ∈ ˚ A i k , ∀ k, i = i = · · · 6 = i n )defines a unital state preserving map from the algebraic free products ( A, φ ) to (
B, ψ ). We denoteby T = ∗ i ∈ I T i , and call it the free product map of the T i . Similarly, we may define the c-free(conditionally free) product state in the sense of Bo˙zejko, Leinert and Speicher [BLS96]. Let( A i , φ i ) be as above and let ρ i be further states respectively on A i for each i . The conditional freeproduct of ( ρ i ) i is the functional ω := ∗ ( ψ i ) ρ i on ( A, φ ) = ∗ i ∈ I ( A i , φ i ) defined by the prescription ω (1) = 1 and ω ( a · · · a n ) = ρ i (1) ( a ) · · · ρ i ( n ) ( a n )for all n ≥ i (1) = · · · 6 = i ( n ) elements in I and a j ∈ ker φ i ( j ) for j = 1 , . . . , n . It is shown in[BLS96, Theorem 2.2] that the conditional free product of states is again a state.1.2. L p -improving operators. Let A be a finite dimensional C*-algebra equipped with a faithfultracial state τ . The associated noncommutative L p -spaces will be denoted by L p ( A ). For a subset E ⊂ A , we denote by E + the positive part of E .Recall that A can be identified with a direct sum of matrix algebras, that is, there exist somefinite dimensional Hilbert spaces H , . . . , H m such that the following ∗ -isomorphism holds A ≃ B ( H ) ⊕ · · · ⊕ B ( H m ) . We will not distinguish the above two C*-algebras in the sequel. For each i ∈ { , . . . , m } , let ξ i , . . . , ξ in i be an orthonormal basis for H i , and define the operator e ipq ∈ B ( H i ) by e ipq ( v ) = SIMENG WANG h v, ξ iq i H i ξ ip for all v ∈ H i and p, q ∈ { , . . . , n i } . Take any x = x ⊕ · · · ⊕ x m ∈ A with x i ∈ B ( H i )for each i ∈ { , . . . , m } , and let λ i , . . . , λ in i be the eigenvalues of | x i | ∈ B ( H i ) (1 ≤ i ≤ m )ranged in non-increasing order and counted according to multiplicity. We can find a direct sum ofunitaries u = u ⊕ u ⊕ · · · ⊕ u m with u i ∈ B ( H i ) for each i such that | x i | ( u i ξ ik ) = λ ik ( u i ξ ik ) for all k ∈ { , . . . , n i } and i ∈ { , . . . , m } , that is, u ∗ | x | u = P i P n i k =1 λ ik e ikk . If we write β ik = τ ( e ikk ) ∈ [0 ,
1] for k ∈ { , . . . , n i } and i ∈ { , . . . , m } , then the L p -norm of x for 1 ≤ p < ∞ is(1.2) k x k pp = τ ( u ∗ | x | p u ) = τ m X i =1 n X k =1 ( λ ik ) p e ikk ! = m X i =1 n X k =1 ( λ ik ) p β ik . We will prove in this section the result below.
Theorem 1.1.
Let A be a finite dimensional C*-algebra equipped with a faithful tracial state τ ,and T : A → A be a unital -positive trace preserving map on A . Then ∃ ≤ p < , k T x k ≤ k x k p , x ∈ A if and only if (1.3) sup x ∈ A \{ } ,τ ( x )=0 k T x k k x k < . Remark . Equivalently we can rewrite the above condition (1.3) assup x ∈ A \{ } ,τ ( x )=0 h| T | x, x ik x k < , which means exactly that the whole eigenspace of | T | for the eigenvalue 1 is just C
1. In this sensewe refer to the above inequality as a spectral gap phenomenon of T .Recall that the L -norms assert some differential properties. The following lemma is elementary. Lemma 1.3.
Let A be a C*-algebra with a state ϕ and T : A → A be a positive map on A . Let O ⊂ A h be an open set in the space A h of all selfadjoint elements in A . The function f : O ∋ x ϕ (( T x ) ) is infinitely (Fr´echet) differentiable in O and for x ∈ O , f ′ ( x ) = ϕ ( T xT · ) + ϕ ( T · T x ) , f ′′ ≡ ϕ ( T · T · ) , f ( n ) ≡ , n ≥ . In general a norm estimate can be reduced to the argument on positive cones.
Lemma 1.4 ([RX16, Remark 9]) . Let M be a von Neumann algebra and T : L p ( M ) → L q ( M ) be a bounded linear map for ≤ p, q ≤ ∞ . Assume that T is -positive in the sense that Id M ⊗ T maps the positive cone of L p ( M ⊗ M ) to that of L q ( M ⊗ M ) . Then k T x k q ≤ k T ( | x | ) k / q k T ( | x ∗ | ) k / q , x ∈ L p ( M ) . Consequently, k T k = sup {k T x k q : x ∈ L p ( M ) + , k x k p ≤ } . Now we give the proof of the theorem.
Proof of the theorem.
Assume firstly 1 ≤ p < k T x k ≤ k x k p for all x ∈ A . Note that k T x k = k| T | x k . Observe that T ∗ is also a positive trace preserving map on A , and hence so is | T | . We choose an element x ∈ A such that τ ( x ) = 0 and | T | x = λx , with λ = sup x ∈ A \{ } ,τ ( x )=0 k T x k k x k = sup x ∈ A \{ } ,τ ( x )=0 k| T | x k k x k . Since | T | is a positive map on A and λ ∈ R , we may assume that x = x ∗ . For any self-adjointelement y ∈ A and 1 ≤ q < ∞ , it is easy to compute that d dε k εy k q (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = ( q − τ ( y ) > . Note also that by assumption k λεx k ≤ k εx k p , ε > . p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 5 Then taking the second derivative at ε = 0 we get λ ≤ ( p − <
1, as desired.Now we suppose (1.3) holds. Set ˚ A = { x ∈ A (cid:12)(cid:12) τ ( x ) = 0 } and take σ = { x ∈ A + (cid:12)(cid:12) τ ( x ) = 1 } =(1 + ˚ A ) + which is exactly the set of positive elements in the unit sphere of L ( A ). We first showthat there exists 1 ≤ p < U of 1 such that(1.4) ∀ x ∈ U ∩ σ, k T x k ≤ k x k p . To begin with, we consider F ( x ) = k T x k − k x k , x ∈ A + . Using the previous lemma we see that F is infinitely differentiable at any x ∈ A + \ { } and F ′ ( x )( y ) = k T x k − τ (( T x )( T y )) − k x k − τ ( xy ) , y ∈ AF ′′ ( x )( y , y ) = −k T x k τ (( T x )( T y )) τ (( T x )( T y )) + k T x k − τ (( T y )( T y ))+ k x k − τ ( xy ) τ ( xy ) − k x k − τ ( y y ) , y , y ∈ A. Since T is unital and preserves the trace, it follows that for y ∈ ˚ A , F ′ (1)( y ) = 0 , F ′′ (1)( y, y ) = k T y k − k y k . Then consider the second order Taylor expansion of F at 1. We can find a δ > k y k ≤ δ , y ∈ ˚ A , we have 1 + y ∈ A + and F (1 + y ) = F (1) + F ′ (1)( y ) + 12 F ′′ (1)( y, y ) + R ( y )= 12 ( k T y k − k y k ) + R ( y ) , R ( y ) = o ( k y k ) . Recall that by (1.3), k T y k − k y k < y ∈ ˚ A . Thus by continuity, c := sup {k T y k − k y k : y ∈ ˚ A, k y k = 1 } < . Since the function y
7→ k
T y k − k y k is 2-homogeneous, we get ∀ y ∈ ˚ A, k T y k − k y k ≤ c k y k . Take δ ∈ (0 , δ ) such that ∀ y ∈ ˚ A, k y k ≤ δ , | R ( y ) |k y k < | c | . Then for y ∈ ˚ A , k y k ≤ δ , ( ∗ ) F (1 + y ) = 12 ( k T y k − k y k ) + R ( y ) ≤ c k y k . On the other hand, consider G ( x ) = k x k − k x k p , x = 1 + y, y = y ∗ ∈ A, k y k < δ . Let y = y ∗ ∈ A with k y k < δ , then by (1.2) we may take some K ∈ N and β , . . . , β K ∈ [0 , L p -norm of x = 1 + y for 1 ≤ p < ∞ is exactly( ∗∗ ) k y k p = K X i =1 β i (1 + λ i ) p ! p where ( λ i ) i ⊂ R is the list of eigenvalues of y . So in order to estimate G , we consider the function g on R K defined as g ( ξ ) = K X i =1 β i (1 + ξ i ) ! − K X i =1 β i (1 + ξ i ) p ! p , ξ = ( ξ , . . . , ξ K ) ∈ R K . A straightforward calculation gives ∂g∂ξ i (0) = 0 , ∂ g∂ξ i ∂ξ j (0) = ( p − β i β j , ∂ g∂ξ i (0) = (2 − p )( β i − β i ) , ≤ i = j ≤ K. SIMENG WANG
So by the Taylor formula g ( ξ ) = 12 X i (2 − p )( β i − β i ) ξ i + 12 X j = k ( p − β j β k ξ j ξ k + R ( ξ ) , R ( ξ ) = o ( k ξ k ) . If 2 − | c | ≤ p ≤ < δ < δ is such that | R ( ξ ) | ≤ | c | P Ki =1 β i ξ i whenever P Ki =1 β i ξ i ≤ δ ,then for any ξ ∈ R K with P Ki =1 β i ξ i ≤ δ , | g ( ξ ) | ≤
12 (2 − p ) K X i =1 ( β i − β i ) ξ i + 12 (2 − p ) K X i =1 β i ξ i + | c | K X i =1 β i ξ i < | c | K X i =1 β i ξ i . This, together with ( ∗∗ ), implies that, putting λ = ( λ , . . . , λ K ), G (1 + y ) = g ( λ ) ≤ | c | K X i =1 β i λ i = | c | k y k , k y k ≤ δ. Combined with ( ∗ ) we deduce k T x k − k x k p = F (1 + y ) + G (1 + y ) ≤ , x = 1 + y, y ∈ ˚ A, k y k ≤ δ, for all p ≥ − | c | := p . So U = { y (cid:12)(cid:12) y = y ∗ ∈ A, k y k < δ } is the desired neighborhood in(1.4).Now we can derive the inequality for all x ∈ σ . For x ∈ σ \ U ⊂ (1 + ˚ A ) + \ { } , we write x = 1 + y with y ∈ ˚ A \{ } and then by (1.3) and the trace preserving property we have k T x k =1 + k T y k < k y k = k x k . Note also that σ is compact, so we can find M < k T x k / k x k < M for all x ∈ σ \ U . Given p <
2, let C p be the optimal constant for the inequality k x k ≤ C p k x k p for x ∈ A , then C p → p →
2. Take p ≥ p such that C p ≤ M − . We getthen ∀ x ∈ σ \ U, k T x k k x k p ≤ M C p ≤ , p ≤ p ≤ . As a result, for all p ∈ [ p , k T x k ≤ k x k p , x ∈ σ. Since the norm is homogeneous and T is 2-positive, the above inequality holds for all x ∈ A aswell. (cid:3) Apart from the above elementary proof, we would like to give an alternative simpler approachwhich yields a little bit stronger conclusion. The argument, however, depends heavily on thefollowing recent and deep result on the convexity of L p -spaces: Theorem 1.5 ([RX16, Theorem 1]) . Let M be a von Neumann algebra equipped with a faithfulsemifinite normal trace φ . Let N be a von Neumann subalgebra such that the restriction of φ to N is semifinite. Denote by E the unique φ -preserving conditional expectation from M onto N .For < p ≤ , we have k x k p ≥ kE x k p + ( p − k x − E x k p , x ∈ L p ( M ) . For < p < ∞ , the inequality is reversed. Immediately we may deduce Theorem 1.1 as follows. Note that the result below is slightlystronger than the statement of Theorem 1.1.
Theorem 1.6.
Let A be a finite dimensional C*-algebra equipped with a faithful tracial state τ ,and T : A → A be a unital trace preserving map on A . Then (1.5) ∃ < p < , ∀ x ∈ A, k T x k ≤ k x k p if and only if λ := sup x ∈ A \{ } ,τ ( x )=0 k T x k k x k < . p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 7 Moreover, if the above assertions are satisfied, then λ ≤ c − p p p − , where c p = sup x ∈ A \{ } ,τ ( x )=0 k x k k x k p . Proof.
The necessity has been already proved in the proof of Theorem 1.1. Now, assume λ < x ∈ A and y = x − τ ( x )1. Write a = τ ( x ). Since T is trace preserving, τ ( T y ) = τ ( y ) = 0. For p ≤ c p the best constant with k · k ≤ c p k · k p . Then ( p − /c p → p → p < p − /c p > λ , then we have k T x k = k a T y k = | a | + k T y k ≤ | a | + λ k y k ≤ | a | + λ c p k y k p ≤ | a | + ( p − k y k p ≤ k x k p , whence (1.5). (cid:3) Remark . Let A be a finite dimensional C*-algebra equipped with a faithful tracial state τ , and T : A → A be a unital trace preserving map on A . Consider the restriction of T on the subspace { x ∈ A : τ ( x ) = 0 } of A and its adjoint, then we see thatsup x ∈ A \{ } ,τ ( x )=0 k T x k k x k = sup x ∈ A \{ } ,τ ( x )=0 k T ∗ x k k x k . Then the above theorem also implies that if there exists 1 < p < ∀ x ∈ A, k T x k ≤ k x k p , then ∀ x ∈ A, k T ∗ x k ≤ k x k p , and equivalently for 2 < q < ∞ with 1 /p + 1 /q = 1, ∀ x ∈ A, k T x k q ≤ k x k . It is easy to see that the free product of unital trace preserving completely positive maps canbe extended to the L p -spaces on using the interpolation between L and L ∞ . But in general itis a delicate problem for the extension of algebraic free product of unital trace preserving mapsonto the associated L p -spaces. Here we provide a method to construct unital trace preserving L p -improving operators on the free product of finite-dimensional C*-algebras. To see this we needthe following trivial claim. Claim . Let M be a finite von Neumann algebra equipped with a faithful tracial state τ . Ifthe vectors e , . . . , e m ∈ M are orthonormal in L ( M , τ ) and denote c = max ≤ k ≤ m k e k k ∞ , thenfor α , . . . , α m ∈ C and 2 ≤ q ≤ ∞ , k m X k =1 α k e k k q ≤ ( cm ) − q k m X k =1 α k e k k . Proof.
Note that k m X k =1 α k e k k ∞ ≤ c / m X k =1 | α k | ≤ c / m / m X k =1 | α k | ! / , which gives the claim for q = ∞ . The inequality for 2 ≤ q ≤ ∞ then follows from the H¨olderinequality. (cid:3) Theorem 1.9.
Let ( A i , τ i ) , ≤ i ≤ n be a finite family of finite dimensional C*-algebras and set ( A , τ ) = ¯ ∗ ≤ i ≤ n ( A i , τ i ) to be the von Neumann algebraic free product. For each ≤ i ≤ n , T i is aunital trace preserving map such that k T i : L p ( A i ) → L ( A i ) k = 1 for some < p < . Then the (algebraic) free product map T = ∗ ≤ i ≤ n T i on ∗ ≤ i ≤ n A i extends toa map such that k T : L p ′ ( A ) → L ( A ) k = 1 SIMENG WANG for some < p ′ < .Proof. By the previous theorem and remark,(1.6) λ = max ≤ i ≤ n sup x ∈ ˚ A i k T i x k k x k = max ≤ i ≤ n sup x ∈ ˚ A i k T ∗ i x k k x k < . Consider R = T ∗ and R i = T ∗ i for all 1 ≤ i ≤ n , then R = R ∗ · · · ∗ R n . By density, consider x ∈ ∗ ≤ i ≤ n ( A i , τ i ) in the algebraic free product and we will show that k Rx k q ≤ k x k for some q > x . Now fix some r ≥
1. For each i , choose afamily ( e ( i ) k ) n i k =1 of eigenvectors of | R i | which forms an orthonormal basis of ˚ A i under τ i , then E r = { e ik = e ( i ) k · · · e ( i r ) k r : 1 ≤ k j ≤ n j , ≤ j ≤ r, i = · · · 6 = i r } forms an orthonormal basis of ⊕ i = ···6 = i r ˚ A i ⊗· · ·⊗ ˚ A i r which are also eigenvectors of | R | . Note that | E r | ≤ n r m r for m = max j n j .Write additionally c = max k,i k e ( i ) k k ∞ . Then for any y r ∈ ⊕ i = ···6 = i r ˚ A i ⊗ · · · ⊗ ˚ A i r the above claimyields(1.7) k y r k q ≤ ( cnm ) r ( − q ) k y r k . Write x = τ ( x )1 + P r ≥ x r where x r ∈ ⊕ i = ···6 = i r ˚ A i ⊗ · · · ⊗ ˚ A i r . Note that k Rx r k ≤ λ r k x r k according to (1.6) and the choice of E r . Together with Theorem 1.5 and (1.7), k Rx k q ≤ | τ ( x ) | + ( q − k X r ≥ Rx r k q ≤ | τ ( x ) | + ( q − X r ≥ k Rx r k q ≤ | τ ( x ) | + ( q − X r ≥ ( cnm ) r ( − q ) k Rx r k ≤ | τ ( x ) | + ( q − X r ≥ ( cnm ) r ( − q ) λ r k x r k . Observe that ( q − cnm ) − q tends to 1 whenever q → λ <
1, so we may choose2 < q < ∞ such that λ ( cnm ) − q ≤ ( q − − . For such a q we then have k Rx k q ≤ | τ ( x ) | + ( q − X r ≥ ( q − − r k x r k ≤ | τ ( x ) | + ( q − X k ≥ ( q − − k X r ≥ k x r k < | τ ( x ) | + X r ≥ k x r k = k x k . Take 1 < p ′ < /p ′ + 1 /q = 1. Then we get k T : L p ′ ( A ) → L ( A ) k = 1. (cid:3) Preliminaries on quantum groups with Fourier analysis
In this section we will do some preparations for discussing convolution operators in the quantumgroup framework. We will start with some preliminaries on compact quantum groups and thenintroduce the Fourier series in this setting. p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 9 Compact quantum groups.
In this short paragraph we recall some basic definitions andproperties of compact quantum groups. All proofs of the facts mentioned below without referencescan be found in [Wor98] and [MVD98].
Definition 2.1.
Consider a unital C*-algebra A and a unital ∗ -homomorphism ∆ : A → A ⊗ A called comultiplication on A such that (∆ ⊗ ι )∆ = ( ι ⊗ ∆)∆ and { ∆( a )(1 ⊗ b ) : a, b ∈ A } and { ∆( a )( b ⊗
1) : a, b ∈ A } are linearly dense in A ⊗ A . Then ( A, ∆) is called a compact quantum group . We denote G = ( A, ∆)and A = C ( G ). We say that G is a finite quantum group if the space A = C ( G ) is finitedimensional.The following fact due to Woronowicz is fundamental in the quantum group theory. Proposition 2.2.
Let G be a compact quantum group. There exists a unique state h on C ( G )( called the Haar state of G ) such that for all x ∈ C ( G ) , ( h ⊗ ι ) ◦ ∆( x ) = h ( x )1 = ( ι ⊗ h ) ◦ ∆( x ) . Let G = ( A, ∆) be a compact quantum group and consider an element u ∈ A ⊗ B ( H ) withdim H = n . We identify A ⊗ B ( H ) = M n ( A ), and write u = [ u ij ] ni,j =1 . Here, u is called an n -dimensional representation of G if for all j, k = 1 , ..., n , we have(2.1) ∆( u jk ) = n X p =1 u jp ⊗ u pk . A representation u is said to be non-degenerate if u is invertible, unitary if u is unitary, and irreducible if the only matrices T ∈ M n ( C ) with T u = uT are multiples of the identity matrix. Tworepresentations u, v ∈ M n ( A ) are called equivalent if there exists an invertible matrix T ∈ M n ( C )such that T u = vT . Denote by Irr( G ) the set of unitary equivalence classes of irreducible unitaryrepresentations of G . For each α ∈ Irr( G ), let u α ∈ C ( G ) ⊗ B ( H α ) be a representative of the class α where H α is the finite dimensional Hilbert space on which u α acts.With the notation above, the ∗ -subalgebra A spanned by { u αij : u α = [ u αij ] n α i,j =1 , α ∈ Irr( G ) } ,usually called the algebra of the polynomials on G , is dense in C ( G ) , and the Haar state h is faithfulon this dense algebra. In the sequel we denote A = Pol( G ). Consider the GNS representation( π h , H h ) of C ( G ), then Pol( G ) can be viewed as a subalgebra of B ( H h ). Define C r ( G ) (resp., L ∞ ( G )) to be the C*-algebra (resp., the von Neumann algebra) generated by Pol( G ) in B ( H h ).Then h extends to a normal faithful state on L ∞ ( G ).It is known that there exists a linear antihomomorphism S on Pol( G ) such that(2.2) S ( S ( a ) ∗ ) ∗ = a, a ∈ Pol( G ) , determined by S ( u αij ) = ( u αji ) ∗ , u α = [ u αij ] n α i,j =1 , α ∈ Irr( G ) .S is called the antipode of G . For a, b ∈ Pol( G ), we have S (( ι ⊗ h )(∆( b )(1 ⊗ a ))) = ( ι ⊗ h )((1 ⊗ b )∆( a )) , (2.3) S (( h ⊗ ι )(( b ⊗ a ))) = ( h ⊗ ι )(∆( b )( a ⊗ . We will use the
Sweedler notation for the comultiplication of an element a ∈ A , i.e. omitthe summation and the index in the formula ∆( a ) = P i a (1) ,i ⊗ a (2) ,i and write simply ∆( a ) = P a (1) ⊗ a (2) .The Peter-Weyl theory for compact groups can be extended to the quantum case. In particular,it is known that for each α ∈ Irr( G ) there exists a positive invertible operator Q α ∈ B ( H α ) suchthat Tr( Q α ) = Tr( Q − α ) := d α and(2.4) h ( u αij ( u βlm ) ∗ ) = δ αβ δ il ( Q α ) mj d α , h (( u αij ) ∗ u βlm ) = δ αβ δ jm ( Q − α ) li d α where β ∈ Irr( G ), 1 ≤ i, j ≤ dim H α , 1 ≤ l, m ≤ dim H β . The dual quantum group ˆ G of G is defined via its “algebra of functions” ℓ ∞ ( ˆ G ) = ⊕ α ∈ Irr( G ) B ( H α )where ⊕ α B ( H α ) refers to the direct sum of B ( H α ), i.e. the bounded families ( x α ) α with each x α in B ( H α ). We will not completely recall the quantum group structure on ˆ G as we do not need itin the following. We only remark that the (left) Haar weight ˆ h on ˆ G can be explicitly given by(see e.g. [VD96, Section 5]) ˆ h : ℓ ∞ ( ˆ G ) ∋ x X α ∈ Irr( G ) d α Tr( Q α p α x ) , where p α is the projection onto H α and Tr denotes the usual trace on B ( H α ) for each α .Our main result will only concentrate on the case where G is of Kac type , that is, its Haar stateis tracial. Proposition 2.3 ([Wor98, Theorem 1.5]) . Let G be a compact quantum group. The Haar state h on C ( G ) is tracial if and only if Q α = Id H α for all α ∈ Irr( G ) in the formula (2.4) and if andonly if the antipode S satisfies S ( x ) = x for all x ∈ Pol( G ) . In particular, if the above conditionsare satisfied and h is faithful on C ( G ) , then S extends to a ∗ -antihomomorphism on C ( G ) whichis positive and bounded of norm one according to (2.2) . Proposition 2.4 ([VD97]) . If G is a finite quantum group, then the Haar state is tracial on C ( G ) . For a compact quantum group G , we write L ( G ) to be the Hilbert space associated to theGNS-construction with respect to the Haar state h . Similarly, denote by ℓ ( ˆ G ) the Hilbert spacegiven by the GNS-construction for ℓ ∞ ( ˆ G ) with respect to the Haar weight ˆ h . If G is of Kac type,for 1 ≤ p ≤ ∞ , we denote additionally by L p ( G ) the L p -space associated to the pair ( L ∞ ( G ) , h ),as defined in the previous subsection.Finally we turn to the dual free product of compact quantum groups. The following constructionis given by [Wan95]. Proposition 2.5.
Let G = ( A, ∆ A ) and G = ( B, ∆ B ) be two compact quantum groups withHaar states h A , h B respectively. There exists a unique comultiplication ∆ on A ∗ c B such that thepair ( A ∗ c B, ∆) forms a compact quantum group, denoted by G = G ˆ ∗ G and we have ∆ | A = ( i A ⊗ i A ) ◦ ∆ A , ∆ | B = ( i B ⊗ i B ) ◦ ∆ B , where i A and i B are the natural embedding of A and B into A ∗ c B respectively. Moreover theHaar state on G is the free product state h A ∗ h B . Fourier analysis.
The Fourier transform for locally compact quantum groups has beendefined in [Kah10], [Coo10] and [Cas13]. In the setting of compact quantum groups, we maygive a more explicit description below. Let a compact quantum group G be fixed. For a linearfunctional ϕ on Pol( G ), we define the Fourier transform ˆ ϕ = ( ˆ ϕ ( α )) α ∈ Irr( G ) ∈ ⊕ α B ( H α ) by(2.5) ˆ ϕ ( α ) = ( ϕ ⊗ ι )(( u α ) ∗ ) ∈ B ( H α ) , α ∈ Irr( G ) . In particular, any x ∈ L ∞ ( G ) (or L ( G )) induces a continuous functional on L ∞ ( G ) defined by y h ( yx ), and the Fourier transform ˆ x = (ˆ x ( α )) α ∈ Irr( G ) of x is given byˆ x ( α ) = ( h ( · x ) ⊗ ι )(( u α ) ∗ ) ∈ B ( H α ) , α ∈ Irr( G ) . The above definition is slightly different from that of [Cas13] or [Kah10]. Indeed, we replace theunitary u α by ( u α ) ∗ in the above formulas. This is just to be compatible with standard definitionsin classical analysis on compact groups such as in [Fol95, Section 5.3], which will not cause essentialdifference. On the other hand, the notation ˆ ϕ has some slight conflict with the dual Haar weightˆ h on ˆ G whereas one can distinguish them by the elements on which it acts, so we hope that thiswill not cause ambiguity for readers.Denote by F : x ˆ x the Fourier transform established above. It is easy to establish the Fourierinversion formula and the Plancherel theorem for L ( G ). As we did not find them explicitly forcompact quantum groups in the literature, we include the detailed calculation of the Fourier seriesin the following proposition. p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 11 Proposition 2.6. (a)
For all x ∈ L ( G ) , we have (2.6) x = X α ∈ Irr( G ) d α ( ι ⊗ Tr)[(1 ⊗ ˆ x ( α ) Q α ) u α ] , where the convergence of the series is in the L -sense. For any α ∈ Irr( G ) , if we denote by E α theorthogonal projection of L ( G ) onto the subspace spanned by by the matrix coefficients ( u αij ) n α i,j =1 ,and write E α x = P i,j x αij u αij with x αij ∈ C , X α = [ x αji ] i,j , then ˆ x ( α ) = d − α X α Q − α . (b) F is a unitary from L ( G ) onto ℓ ( ˆ G ) .Proof. (a) Denote by E α the subspace spanned by the matrix units ( u αij ) n α i,j =1 for α ∈ Irr( G ). ThenPol( G ) is spanned by all the E α , α ∈ Irr( G ). It is easy to see from H¨older’s inequality that Pol( G )is k · k -dense in L ∞ ( G ), and also recall that L ( G ) is the k · k -completion of L ∞ ( G ), so Pol( G ) is k · k -dense in L ( G ) and L ( G ) is a Hilbert direct sum of the orthogonal subspaces ( E α ) α ∈ Irr( G ) .So each x ∈ L ( G ) can be written as(2.7) x = X α ∈ Irr( G ) E α x = X α ∈ Irr( G ) X i,j x αij u αij , ( x αij ∈ C )where E α x := P i,j x αij u αij is the orthogonal projection of x onto E α .Now for α ∈ Irr( G ), write u α = P l,m u αlm ⊗ e αlm and X α = [ x αji ] i,j where e αlm denote the canonicalmatrix units of B ( H α ). Thenˆ x ( α ) = ( h ( · x ) ⊗ ι )(( u α ) ∗ ) = ( h ( ·E α x ) ⊗ ι )(( u α ) ∗ ) + ( h ( ·E ⊥ α x ) ⊗ ι )(( u α ) ∗ )(2.8) = X i,j,l,m x αij (cid:0) h ( · u αij ) ⊗ ι (cid:1) (( u αlm ) ∗ ⊗ e αml ) + 0= X i,j,l,m x αij h (cid:0) ( u αlm ) ∗ u αij (cid:1) e αml = d − α X i,j,l x αij ( Q − α ) il e αjl = d − α X α Q − α . Hence d α ( ι ⊗ Tr)[(1 ⊗ ˆ x ( α ) Q α ) u α ] = X i,j,l,m ( ι ⊗ Tr)[(1 ⊗ x αji e αij )( u αlm ⊗ e αlm )]= X i,j,m x αji ( ι ⊗ Tr)( u αjm ⊗ e αim ) = X i,j x αji u αji = E α x. Combining the last equality with (2.7) proves the desired (2.6).(b) Let x = X α ∈ Irr( G ) E α x = X α ∈ Irr( G ) X i,j x αij u αij ∈ L ( G ) . For each α ∈ Irr( G ), kE α x k = h (( E α x ) ∗ ( E α x )) = X i,j,l,m x αij x αlm h (cid:0) ( u αij ) ∗ u αlm (cid:1) = d − α X i,j,l x αij x αlj ( Q − α ) li and also by (2.8) d α Tr( Q α ˆ x ( α ) ∗ ˆ x ( α )) = Tr( X ∗ α X α Q − α ) = X i,j,l x αij x αlj ( Q − α ) li . Hence by Parseval’s identity, k x k = X α kE α x k = X α d − α X i,j,l x αij x αlj ( Q − α ) li = X α d − α d α Tr( Q α ˆ x ( α ) ∗ ˆ x ( α )) = k ˆ x k . Thus F maps isometrically L ( G ) into ℓ ( ˆ G ). From (2.6) and the isometric relation we see thatthe range of F contains the subset of all finitely supported families ( a α ) ∈ ⊕ α B ( H α ), which isdense in ℓ ( ˆ G ). Therefore F is surjective and hence unitary. (cid:3) Example 2.7. (1) Let G be a compact group and define∆( f )( s, t ) = f ( st ) , f ∈ C ( G ) , s, t ∈ G. Then G = ( C ( G ) , ∆) is a compact quantum group. The elements in Irr( G ) := Irr( G ) coincide withthe usual strongly continuous irreducible unitary representations of G . Any continuous functional ϕ on C ( G ) corresponds to a complex Radon measure µ on G by the Riesz representation theorem.By definition (2.5), the Fourier series of µ is given byˆ µ ( π ) = ( ϕ ⊗ ι )( π ( · ) ∗ ) = Z G π ( g ) ∗ dµ ( g ) , π ∈ Irr( G ) . In particular for f ∈ L ( G ), we haveˆ f ( π ) = Z G π ( g ) ∗ f ( g ) dg, π ∈ Irr( G )and we have the Fourier expansion and the Plancherel formula f = X π ∈ Irr( G ) d π Tr( ˆ f ( π ) π ) , k f k = X π ∈ Irr( G ) d π k ˆ f ( π ) k where d π is the dimension of the Hilbert space on which the representation π acts and kk HS denotesthe usual Hilbert-Schmidt norm. We refer to [Fol95, Section 5.3] and [HR70, pp.77-87] for moreinformation.(2) Let Γ be a discrete group with its neutral element e and C ∗ r (Γ) be the associated reducedgroup C*-algebra generated by λ (Γ) ⊂ B ( ℓ (Γ)), where λ denotes the left regular representation.The “dual” G = ˆΓ of Γ is a compact quantum group such that C ( G ) is the group C*-algebra C ∗ r (Γ) equipped with the comultiplication ∆ : C ∗ r (Γ) → C ∗ r (Γ) ⊗ C ∗ r (Γ) defined by∆( λ ( γ )) = λ ( γ ) ⊗ λ ( γ ) , γ ∈ Γ . The Haar state of G is the unique trace τ on C ∗ r (Γ) such that τ (1) = 1 and τ ( λ ( γ )) = 0 for γ ∈ Γ \ { e } . The elements of λ (Γ) give all irreducible unitary representations of G , which are allof dimension 1. It is easy to check from definition that for any f ∈ C ∗ r (Γ),ˆ f ( γ ) = τ ( f λ ( γ ) ∗ ) , γ ∈ Γ . And any f ∈ L ( G ) has an expansion such that f = X γ ∈ Γ ˆ f ( γ ) λ ( γ ) , k f k = X γ ∈ Γ | ˆ f ( γ ) | . Let us end the section with a brief description of multipliers and convolutions in terms of Fourierseries. Return back to a general compact quantum group G . For a = ( a α ) α ∈ ⊕ α B ( H α ), we definethe left multiplier m a : Pol( G ) → Pol( G ) associated to a by(2.9) m a x = X α ∈ Irr( G ) d α ( ι ⊗ Tr)[(1 ⊗ ˆ x ( α ) a α Q α ) u α ] , x ∈ Pol( G ) , which means that(2.10) ( m a x )ˆ( α ) = ˆ x ( α ) a α , α ∈ Irr( G ) . p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 13 In the same way we may define the right multiplier m ′ a : Pol( G ) → Pol( G ) such that m ′ a x = X α ∈ Irr( G ) d α ( ι ⊗ Tr)[(1 ⊗ a α ˆ x ( α ) Q α ) u α ] , x ∈ Pol( G ) . Observe that the multiplier m a (or m ′ a resp.) is unital, i.e. m a (1) = 1 ( m ′ a (1) = 1 resp.) if andonly if a = 1. Remark . In case G is of Kac type, that is, Q α = Id H α for all α ∈ Irr( G ) by Proposition 2.3,the multipliers m a and m ′ a can be equivalently defined by(2.11) ( m a ⊗ ι )( u α ) = (1 ⊗ a α )( u α ) , ( m ′ a ⊗ ι )( u α ) = ( u α )(1 ⊗ a α ) , α ∈ Irr( G ) , which corresponds to the standard definition of left and right multipliers on locally compactquantum groups in [JNR09] and [Daw12]. If G is not of Kac type, the above formula (2.11) givesa similar equality corresponding to (2.10), that is, ( m a x )ˆ( α ) = ˆ x ( α ) Q α a α Q − α , α ∈ Irr( G ).We will use the standard definition of convolution products given by Woronowicz. Let x ∈ C ( G )and ϕ, ϕ ′ be linear functionals on C ( G ). We define ϕ ⋆ ϕ ′ = ( ϕ ⊗ ϕ ′ ) ◦ ∆ ,x ⋆ ϕ = ( ϕ ⊗ ι )∆( x ) ,ϕ ⋆ x = ( ι ⊗ ϕ )∆( x ) . Observing the embedding x h ( · x ) from C ( G ) into C ( G ) ∗ , the convolution products definedabove are related as follows according to (2.3) (see also [VD07, Proposition 2.2]): on Pol( G ) wehave(2.12) h ( · x ) ⋆ ϕ = h ( · [( ϕ ◦ S ) ⋆ x ]) , ϕ ⋆ h ( · x ) = h ( · [ x ⋆ ( ϕ ◦ S − )]) . We note that for α ∈ Irr( G ) and u α = [ u αij ] ≤ i,j ≤ n α , (cid:2) ϕ (( u αji ) ∗ ) (cid:3) i,j = ( ϕ ⊗ ι )(( u α ) ∗ ) = ˆ ϕ ( α ) . Then by (2.1), a straightforward calculation shows that(2.13) ( ϕ ⋆ ϕ ′ )ˆ( α ) = ˆ ϕ ′ ( α ) ˆ ϕ ( α ) . Hence together with (2.12),(2.14) ( x ⋆ ϕ )ˆ( α ) = ˆ x ( α )( ϕ ◦ S )ˆ( α ) , ( ϕ ⋆ x )ˆ( α ) = ( ϕ ◦ S − )ˆ( α )ˆ x ( α ) . Non-degenerate states and applications to Hopf images
In this short section we give the key lemma on non-degenerate states, which will be of use forour main results. We need the following observation adapted from [Wor98, Lemma 2.1]. Theresult is mentioned in [So l05].
Lemma 3.1.
Let G be a compact quantum group and A = C ( G ) . Suppose that ( ρ i ) i ∈ I is a familyof states on A separating the points of A + , i.e., ∀ x ∈ A + \{ } , ∃ i ∈ I , ρ i ( x ) > . If ρ is a stateon A such that ∀ i ∈ I, ρ ⋆ ρ i = ρ i ⋆ ρ = ρ, then ρ is the Haar state h of G .Proof. Set I = { q ∈ A ⊗ A : ∀ i ∈ I, ( ρ i ⊗ ρ )( q ∗ q ) = 0 } . Then I is a closed left ideal of A ⊗ A . DefineΨ L ( x ) = ( ρ ⊗ ι )∆( x ) − ρ ( x )1 , x ∈ A. Since Ψ L is a difference of two unital completely positive maps, we see that Ψ L is a completelybounded map with norm at most 2. We will prove that(Ψ L ⊗ ι )∆( A ) ⊂ I . In fact, given x ∈ A , by the coassociativity of ∆ we have q := (Ψ L ⊗ ι )∆( x ) = ( ρ ⊗ ι ⊗ ι )( ι ⊗ ∆)∆( x ) − ⊗ [( ρ ⊗ ι )∆( x )]= ∆(( ρ ⊗ ι )∆( x )) − ⊗ [( ρ ⊗ ι )∆( x )] . Thus q ∗ q = ∆ ([( ρ ⊗ ι )∆( x )] ∗ ( ρ ⊗ ι )∆( x )) − ∆(( ρ ⊗ ι )∆( x )) ∗ [1 ⊗ ( ρ ⊗ ι )∆( x )] − [1 ⊗ (( ρ ⊗ ι )∆( x )) ∗ ]∆(( ρ ⊗ ι )∆( x )) + 1 ⊗ ([( ρ ⊗ ι )∆( x )] ∗ [( ρ ⊗ ι )∆( x )])and hence for any i ∈ I we may write( ρ i ⊗ ρ )( q ∗ q ) = q − q − q + q where by the convolution invariance assumption and the coassociativity of ∆ we have q = ( ρ i ⊗ ρ )∆ ([( ρ ⊗ ι )∆( x )] ∗ ( ρ ⊗ ι )∆( x )) = ρ ([( ρ ⊗ ι )∆( x )] ∗ ( ρ ⊗ ι )∆( x )) ,q = q ∗ ,q = ( ρ i ⊗ ρ ) (cid:0) [1 ⊗ (( ρ ⊗ ι )∆( x )) ∗ ]∆(( ρ ⊗ ι )∆( x )) (cid:1) = ρ (cid:0) [(( ρ ⊗ ι )∆( x )) ∗ ]( ρ i ⊗ ι )(( ρ ⊗ ι ⊗ ι )( ι ⊗ ∆)∆( x )) (cid:1) = ρ (cid:0) [(( ρ ⊗ ι )∆( x )) ∗ ]( ρ ⊗ ρ i ⊗ ι )((∆ ⊗ ι )∆( x )) (cid:1) = ρ ([( ρ ⊗ ι )∆( x )] ∗ ( ρ ⊗ ι )∆( x )) ,q = ( ρ i ⊗ ρ ) (cid:0) ⊗ [(( ρ ⊗ ι )∆( x )) ∗ ( ρ ⊗ ι )∆( x )] (cid:1) = ρ ([( ρ ⊗ ι )∆( x )] ∗ ( ρ ⊗ ι )∆( x )) . Note that q = q = q = q . So ( ρ i ⊗ ρ )( q ∗ q ) = 0 and (Ψ L ⊗ ι )∆( A ) ⊂ I is proved.Now by the density of (1 ⊗ A )∆( A ) in A ⊗ A and the complete boundedness of Ψ L , it followsthat Ψ L ( A ) ⊗ ⊂ (1 ⊗ A )(Ψ L ⊗ ι )∆( A ) is also contained in the closed left ideal I , which meansthat for any i ∈ I and x ∈ A , ρ i (Ψ L ( x ) ∗ Ψ L ( x )) = ρ i ⊗ ρ (Ψ L ( x ) ∗ Ψ L ( x ) ⊗
1) = 0 . Recall that ( ρ i ) i ∈ I separates the points of A + , so we have Ψ L ( x ) = 0 and ( ρ ⊗ ι )∆( x ) = ρ ( x )1 forall x ∈ A .A similar argument applies as well to the map Ψ R ( x ) = ( ι ⊗ ρ )∆( x ) − ρ ( x )1, x ∈ A . So ρ = h is the Haar state. (cid:3) Remark . We remark that in case G is a finite quantum group, we can provide a simpler proofof the above lemma. Indeed, since C ( G ) is finite-dimensional, its dual space C ( G ) ∗ is also finite-dimensional, and we can take a maximal linear independent family { ρ i , . . . , ρ i s } ⊂ ( ρ i ) i ∈ I whichform a basis of the subspace spanned by ( ρ i ) i ∈ I in C ( G ) ∗ . Given a nonzero x ∈ A + , there is an i ∈ I such that ρ i ( x ) >
0. Write ρ i = P sk =1 a k ρ i k ( a k ∈ C ), then we see clearly that there existsat least one k ∈ { , . . . , s } such that ρ i k ( x ) = 0 in order that ρ i ( x ) >
0. Then ρ ′ = s P sk =1 ρ i k isfaithful on C ( G ) and ρ ⋆ ρ ′ = ρ ′ ⋆ ρ = ρ , thus ρ is the Haar state by [Wor98, Lemma 2.1].We immediately obtain the following fact. Lemma 3.3.
Let G be a compact quantum group and ϕ be a state on C ( G ) . If ϕ is non-degenerate on C ( G ) in the sense that for all nonzero x ∈ C ( G ) + there exists k ≥ such that ϕ ⋆k ( x ) > ,then w ∗ - lim n →∞ n n X k =1 ϕ ⋆k = h. If additionally h is faithful on C ( G ) , then the converse also holds.Proof. If the above limit holds and h is faithful on C ( G ), then clearly ϕ is non-degeneratesince if there existed a nonzero x ≥ ϕ ⋆n ( x ) = 0 for all n , then we would havelim n n P nk =1 ϕ ⋆k ( x ) = 0, which contradicts the faithfulness of h . On the other hand, if ϕ is non-degenerate, the family of states { n P nk =1 ϕ ⋆k : n ≥ } separates the points of C ( G ) + , so anyaccumulation point of this family becomes the unique Haar state by our previous lemma. (cid:3) p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 15 We would like to digress momentarily to see an application of above lemmas to some prob-lems concerning Hopf images. Let A be a unital C*-algebra and π : C ( G ) → A be a unital ∗ -homomorphism. The Hopf image of π , firstly introduced by Banica and Bichon in [BB10], isthe largest algebra C ( G π ) for some compact quantum subgroup G π ⊂ G such that π factorizesthrough C ( G π ). In this paper we will use the following equivalent characterization recently givenin [SS15]: for each k consider π k = ( π ⊗ · · · ⊗ π | {z } k ) ◦ ∆ ( k − : C ( G ) → A ⊗ k and let I = ∩ ∞ k =1 ker π k . Then, there is a compact quantum group G π = ( B, ∆ π ) with a ∗ -homomorphism π q : B → A such that(3.1) B = C ( G ) / I , ∆ π ◦ q = ( q ⊗ q ) ◦ ∆ , π = π q ◦ q where q : C ( G ) → C ( G ) / I denotes the quotient map. The algebra B = C ( G π ) is exactly the Hopfimage of π . Now let h G , h G π be the Haar states on G , G π respectively. A related question raisedin [BFS12] is the computation of the associated idempotent state h G π ◦ q on G . Simply based onLemma 3.3, the following property generalizes the main result of [BFS12]. Theorem 3.4.
Let G be a compact quantum group and A be a unital C*-algebra with a unital ∗ -homomorphism π : C ( G ) → A . Let G π be the compact quantum group constructed above. Thengiven any faithful state ϕ on A , h G π ◦ q = w ∗ - lim n →∞ n n X k =1 ( ϕ ◦ π ) ⋆k . Proof.
Let ϕ be a faithful state on A and let I = ∩ ∞ k =1 ker π k with π k , k ≥ ϕ ◦ π q is non-degenerate on B = C ( G π ). Consider any x = q ( y ) with y ∈ C ( G ) + satisfying ∀ k ≥ , ( ϕ ◦ π q ) ⋆k ( x ) = 0 . Since, by (3.1),( ϕ ◦ π q ) ⋆k = [( ϕ ⊗ · · · ⊗ ϕ ) ◦ ( π q ⊗ ◦ ⊗ π q )]∆ ( k − ( x )= [( ϕ ⊗ · · · ⊗ ϕ ) ◦ (( π q ◦ q ) ⊗ · · · ⊗ ( π q ◦ q ))]∆ ( k − ( y ) = ( ϕ ⊗ · · · ⊗ ϕ )( π k ( y ))and since ϕ is faithful, we get y ∈ ∩ ∞ k =1 ker π k , which means that x = q ( y ) = 0. As a result ϕ ◦ π q is non-degenerate. Therefore we have h G π = w ∗ - lim n →∞ n P nk =1 ( ϕ ◦ π q ) ⋆k , and hence using (3.1)again, h G π ◦ q = w ∗ - lim n →∞ n n X k =1 ( ϕ ◦ π q ) ⋆k ◦ q = w ∗ - lim n →∞ n n X k =1 ( ϕ ◦ π ) ⋆k , as desired. (cid:3) Main results
In this section we aim to give several characterizations of L p -improving convolutions given bystates on finite quantum groups, and also give the constructions for the free product of finitequantum groups. We will start with some discussions on multipliers on compact quantum groups.In this section we keep the notation of multipliers m a , m ′ a and convolutions ϕ ⋆ϕ given in Section2. Lemma 4.1.
Let G be a compact quantum group of Kac type. Suppose a ∈ ℓ ∞ ( ˆ G ) such that m a (resp. m ′ a ) extends to a unital left (resp., right) multiplier on L ( G ) and b = aa ∗ . Then lim n n P nk =1 m kb x = h ( x )1 for all x ∈ L ( G ) if and only if lim n n P nk =1 m ′ kb x = h ( x )1 for all x ∈ L ( G ) if and only if k a α k < for all α ∈ Irr( G ) \ { } . Proof.
Without loss of generality we only discuss the left multiplier m a . Assume that k a α k < α ∈ Irr( G ) \ { } . By the Plancherel theorem 2.6 and the formula (2.9) we note that m b extends to a bounded map of norm one on L ( G ). We first consider the case x ∈ Pol( G ), so thatˆ x is finitely supported. Let α ∈ Irr( G ) \ { } and k a α k <
1. Then k ( 1 n n X k =1 m kb x )ˆ( α ) k = k n n X k =1 ˆ x ( α )( a α a ∗ α ) k k ≤ n n X k =1 k a α k k k ˆ x ( α ) k → n → ∞ . And for α = 1,( 1 n n X k =1 m kb x )ˆ(1) = ˆ x (1) = h ( x ) . Thus by the Plancherel theorem k n n X k =1 m kb x − h ( x )1 k = X α =1 d α k ( 1 n n X k =1 m kb x )ˆ( α ) k → n → ∞ . Since n P nk =1 m kb is a contraction on L ( G ) and Pol( G ) is dense in L ( G ), we getthe convergence lim n n P nk =1 m kb x = h ( x )1 for all x ∈ L ( G ).Conversely, if ∃ α ∈ Irr( G ) \{ } , k a α k = 1, then viewing b α as a matrix in M n α , we observethat 1 ∈ σ ( b α ) and there exists a nonzero x α ∈ M n α such that x α b α = x α . Take x ∈ L ( G )such that ˆ x (1) = 1, ˆ x ( α ) = x α , ˆ x ( α ) = 0 for α ∈ Irr( G ) \{ , α } . Then m b x = x and hence n P nk =1 m kb x = x does not converge to h ( x )1. (cid:3) Remark . In case the compact quantum group G is not of Kac type, the above argument stillremains true for right multipliers.The following first main result is now in reach. We will consider the case where G is a finitequantum group. Theorem 4.3.
Let G be a finite quantum group. Suppose a ∈ ℓ ∞ ( ˆ G ) is such that m a (resp., m ′ a )is a unital left (resp., right) multiplier on C ( G ) . Then the following assertions are equivalent: (1) there exists ≤ p < such that, ∀ x ∈ C ( G ) , k m a x k ≤ k x k p ;(2) there exists ≤ p < such that, ∀ x ∈ C ( G ) , k m ′ a x k ≤ k x k p ;(3) k a α k < for all α ∈ Irr( G ) \ { } ; (4) lim n n P nk =1 m kb x = h ( x )1 for all x ∈ C ( G ) when b = aa ∗ ; (5) lim n n P nk =1 m ′ kb x = h ( x )1 for all x ∈ C ( G ) when b = aa ∗ .Proof. Without loss of generality we only discuss the left multiplier m a and prove the equivalence(1) ⇔ (3) ⇔ (4).It is easy to see from Plancherel’s theorem that (3) is just (1.3) for T = m a . In fact note thatfor x ∈ C ( G ), h ( x ) = 0 if and only if ˆ x (1) = 0, so (3) implies (1.3) via Plancherel’s theorem. Onthe other hand, suppose by contradiction that there exists α ∈ Irr( G ) \ { } such that k a α k = 1.By Proposition 2.6 we may take a nonzero x ∈ C ( G ) such that ˆ x (1) = 0, ˆ x ( α ) = 0 when k a α k < k ˆ x ( α ) a α k = k ˆ x ( α ) k when k a α k = 1. Then k m a x k = k x k with h ( x ) = 0. As a consequencethe equivalence (1) ⇔ (3) follows from Theorem 1.6.The equivalence between (3) and (4) was proved in the previous lemma. Therefore the theoremis established. (cid:3) Now we turn to the corresponding convolution problems. Let ϕ ∈ C ( G ) ∗ for a compact quantumgroup G . Recall the formula (2.13), and then we have(4.1) (cid:2) ϕ ⋆n (( u αji ) ∗ ) (cid:3) = ( ϕ ⋆n )ˆ( α ) = ˆ ϕ ( α ) n . p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 17 Note that the convergence n P nk =1 m k ˆ ϕ ∗ ( x ) → h ( x )1 for all x ∈ Pol( G ), by (2.10) can be reformu-lated in terms of Fourier coefficients as n n X k =1 m k ˆ ϕ ∗ ( x ) ! ˆ(1) = h ( x ) ˆ ϕ (1) ∗ = h ( x )1 , lim n n n X k =1 m k ˆ ϕ ∗ ( x ) ! ˆ( α ) = lim n n n X k =1 ( ˆ ϕ ( α ) ∗ ) k ˆ x ( α ) = 0 , α ∈ Irr( G ) \{ } . This is to say, ˆ ϕ (1) = 1 , lim n n n X k =1 ˆ ϕ ( α ) k = 0 , α ∈ Irr( G ) \{ } , which, according to (4.1), is equivalent to n P nk =1 ϕ ⋆k ( u αij ) → h ( u αij ) for all α ∈ Irr( G ) and1 ≤ i, j ≤ n α , or in other words,1 n n X k =1 ϕ ⋆k ( x ) → h ( x ) , n → ∞ , x ∈ Pol( G ) . Any state ϕ on C ( G ) induces two convolution operators on C ( G ) T ϕ : C ( G ) ∋ x x ⋆ ϕ = ( ϕ ⊗ ι )∆( x ) , T ′ ϕ : C ( G ) ∋ x ϕ ⋆ x = ( ι ⊗ ϕ )∆( x ) . If additionally G is of Kac type and the Haar state is faithful on C ( G ), then by Proposition 2.3 theantipode S extends to a positive linear operator on C ( G ) and S = S − , and hence ϕ ◦ S = ϕ ◦ S − is also a state. In this case we have h ϕ ( u αji ) i i,j = ˆ ϕ ( α ) , ( ϕ ◦ S )ˆ( α ) = (cid:2) ϕ ( u αij ) (cid:3) i,j = ˆ ϕ ( α ) ∗ and by (2.14) we have ( x ⋆ ϕ )ˆ( α ) = ˆ x ( α ) ˆ ϕ ( α ) ∗ and ( ϕ ⋆ x )ˆ( α ) = ˆ ϕ ( α ) ∗ ˆ x ( α ) for α ∈ Irr( G ), x ∈ C ( G ). So T ϕ = m ˆ ϕ ∗ and T ′ ϕ = m ′ ˆ ϕ ∗ are unital completely positive left and right multipliers,respectively.Now with these remarks and Lemma 3.3 in hand, we may reformulate Theorem 4.3 in terms ofconvolution operators using the above arguments. Theorem 4.4.
Let G be a finite quantum group and ϕ be a state on C ( G ) . Denote ψ = ( ϕ ◦ S ) ⋆ ϕ .The following assertions are equivalent: (1) there exists ≤ p < such that, ∀ x ∈ C ( G ) , k ϕ ⋆ x k ≤ k x k p ;(2) there exists ≤ p < such that, ∀ x ∈ C ( G ) , k x ⋆ ϕ k ≤ k x k p ;(3) k ˆ ϕ ( α ) k < for all α ∈ Irr( G ) \ { } ; (4) lim n n P nk =1 ψ ⋆k = h ; (5) For any nonzero x ∈ C ( G ) + , there exists n ≥ such that ψ ⋆n ( x ) > . Note that if C ( G ) is commutative, i.e. C ( G ) = C ( G ) where G is a finite group, then ϕ ∈ C ( G ) ∗ corresponds to a Radon measure µ via the Riesz representation theorem. The above condition(5) in the theorem just asserts that G is the union of D n := supp ( ν ⋆n ), n ≥
1, where ν denotesthe Radon measure corresponding to ψ . It is easy to see that D = (cid:8) i − j : i, j ∈ supp ( µ ) (cid:9) and D n = D n . So the above corollary covers Ritter’s result [Rit84]. Corollary 4.5.
Let G be a finite group and µ be a probability measure on G . Then there is a ≤ p < such that k x ⋆ µ k ≤ k x k p , x ∈ L p ( G ) if and only if G is equal to the subgroup generated by (cid:8) i − j : i, j ∈ supp ( µ ) (cid:9) . On the other hand, let Γ be a finite group with neutral element e and C ∗ (Γ) be the associatedgroup C*-algebra generated by λ (Γ) ⊂ B ( ℓ (Γ)), where λ denotes the left regular representation.Recall that if G = ˆΓ, then C ( G ) is the group C*-algebra C ∗ (Γ) equipped with the comultiplication∆ : C ∗ (Γ) → C ∗ (Γ) ⊗ C ∗ (Γ) defined by∆( λ ( γ )) = λ ( γ ) ⊗ λ ( γ ) , γ ∈ Γ . Note that any state Φ on C ( G ) corresponds to a positive definite function ϕ on Γ with ϕ ( e ) = 1via the relation Φ( λ ( γ )) = ϕ ( γ ) for all γ ∈ Γ. Therefore we haveΦ ⋆ λ ( γ ) = λ ( γ ) ⋆ Φ = (Φ ⊗ ι )∆( λ ( γ )) = (Φ ⊗ ι )( λ ( γ ) ⊗ λ ( γ )) = ϕ ( γ ) λ ( γ ) , so the convolution operators associated to Φ are just the Fourier-Schur multiplier on Γ associatedto ϕ . Our preceding argument in particular yields the following result extending [Rit84, Theorem2(a)]. Corollary 4.6.
Let Γ be a finite group and ϕ be a positive definite function on Γ with ϕ ( e ) = 1 .Let M ϕ be the associated Fourier-Schur multiplier operator determined by M ϕ ( λ ( γ )) = ϕ ( γ ) λ ( γ ) for all γ ∈ Γ . Then there exists ≤ p < such that k M ϕ x k ≤ k x k p , x ∈ C ∗ (Γ) if and only if | ϕ ( γ ) | < for any γ ∈ Γ \ { e } .Remark . We remark that the finite-dimensional condition cannot be removed in any of ourresults, including Theorem 1.1, Theorem 4.3, Corollary 4.4-4.6. Here we give a counterexampleillustrating this. Let T be the unit circle in the complex plane, then T gives an infinite compactquantum group. Define an operator T : C ( T ) → C ( T ) by T ( f ) = (1 − λ ) τ ( f ) + λf, f ∈ C ( T ) , where 0 < λ < τ denote the usual integral against the normalized Lebesgue measure on T . Then T is obviously unital completely positive since so are τ and the identity map. It is aleft multiplier satisfying T ( f )ˆ(0) = ˆ f (0) and T ( f )ˆ( n ) = λ ˆ f ( n ) for n = 0. Also we may view T as a convolution operator associated to the state f (1 − λ ) τ ( f ) + λf (1) on C ( T ), which isfaithful since τ is faithful. Note that T admits the spectral gap inequality (1.3) as well, and infact, k T f k = λ k f k < k f k for any f ∈ C ( T ) with τ ( f ) = 0. However, there doesn’t exist any p < k T f k ≤ k f k p for all f ∈ C ( T ). Indeed if such a p existed, then for any f ∈ C ( T ),we would have k f k ≥ k f k p ≥ k T f k = τ ( f ) + λ k f − τ ( f ) k ≥ λ ( τ ( f ) + k f − τ ( f ) k ) = λ k f k , which yields an impossible equivalence between the norms k · k and k · k p .In spite of the above general remark, Theorem 1.9 still gives constructions of L p -improvingpositive convolution operators on infinite compact quantum groups. Let G , . . . , G n be finitequantum groups with Haar states h , . . . , h n respectively and let each ϕ i be a state on C ( G i ), i ∈ { , . . . , n } . Denote G = G ˆ ∗ · · · ˆ ∗ G n with the Haar state h and consider the convolutionoperators T i : x x ⋆ ϕ i , x ∈ C ( G i ). Note that the free product map T = ∗ ≤ i ≤ n T i on C ( G ) isjust the convolution operator given by the c-free product state ϕ = ∗ ( h ,...,h n ) ϕ i , i.e.,(4.2) T ( x ) = ( ϕ ⊗ ι )∆( x ) , x ∈ C ( G ) . In fact, we note that if h ( a ) = 0, then h ( a (1) ) = h ( a (2) ) = 0 by (2.1), where ∆( a ) := P a (1) ⊗ a (2) denotes the Sweedler notation. Now for a reduced word x = x · · · x m with x k ∈ C ( G i k ) such that h ( x k ) = 0, i = · · · 6 = i n , i k ∈ { , . . . , n } for each k = 1 , . . . , m ,, we have T ( x ) = T i ( x ) · · · T i m ( x m ) = X ϕ i ( x ) x · · · X ϕ i m ( x m (1) ) x m (2) = X ϕ ( x (1) ) x (2) = ( ϕ ⊗ ι )∆( x ) p -IMPROVING CONVOLUTION OPERATORS ON FINITE QUANTUM GROUPS 19 where we have used the fact that the comultiplication ∆ is an homomorphism. Then the equality(4.2) follows from a standard density argument. Now taking in Theorem 1.9 each T i to be aconvolution operator on a finite quantum group, we get the following corollary: Corollary 4.8.
Let G , . . . , G n be finite quantum groups and let each ϕ i be a state on C ( G i ) , i ∈ { , . . . , n } . Denote G = G ˆ ∗ · · · ˆ ∗ G n and ϕ = ∗ ( h ,...,h n ) ϕ i . If each ϕ i satisfies any oneof the conditions (1)-(5) in Corollary 4.4, then the free product convolution operator given by T : x x ⋆ ϕ, x ∈ C ( G ) is a unital left multiplier on G satisfies k T : L p ( G ) → L ( G ) k = 1 for a certain < p < . Example 4.9.
Now we give a simple method to create nontrivial L p -improving positive convo-lutions (i.e. the associated state is different from the Haar state) on finite and infinite compactquantum groups. Let G be a finite quantum group and h the Haar state on it. Given any state ϕ on C ( G ) and any 0 < λ <
1, we can define a state ρ on C ( G ) by ρ = λϕ + (1 − λ ) h. This is a faithful state which is in particular non-degenerate, and hence by Theorem 4.4 theconvolution operator T ρ : x x ⋆ ρ, x ∈ C ( G ) satisfies k T ρ : L p ( G ) → L ( G ) k = 1for a certain 1 < p < T ρ ′ : x x ⋆ ρ ′ , x ∈ C ( G ˆ ∗ G ) given by the c-free product state ρ ′ = ρ ∗ ( h,h ) ρ satisfies k T ρ ′ : L p ′ ( G ˆ ∗ G ) → L ( G ˆ ∗ G ) k = 1for some 1 < p ′ < Acknowledgment.
The author is indebted to his advisors Quanhua Xu and Adam Skalski fortheir helpful discussions and constant encouragement, and to Professor Gilles Pisier for his carefulreading and pointing out a mistake in the preprint version. The author also thanks the referee fora careful reading of the manuscript and useful suggestions. This research was partially supportedby the NCN (National Centre of Science), grant no. 2014/14/E/ST1/00525.
References [BB10] T. Banica and J. Bichon. Hopf images and inner faithful representations.
Glasg. Math. J. , 52(3):677–703,2010.[BFS12] T. Banica, U. Franz, and A. Skalski. Idempotent states and the inner linearity property.
Bull. Pol. Acad.Sci. Math. , 60(2):123–132, 2012.[BLS96] M. Bo˙zejko, M. Leinert, and R. Speicher. Convolution and limit theorems for conditionally free randomvariables.
Pacific J. Math. , 175(2):357–388, 1996.[Cas13] M. Caspers. The L p -Fourier transform on locally compact quantum groups. J. Operator Theory ,69(1):161–193, 2013.[Coo10] T. Cooney. A Hausdorff-Young inequality for locally compact quantum groups.
Internat. J. Math. ,21(12):1619–1632, 2010.[Daw12] M. Daws. Completely positive multipliers of quantum groups.
Internat. J. Math. , 23(12):1250132, 23,2012.[Fol95] G. B. Folland.
A course in abstract harmonic analysis . Studies in Advanced Mathematics. CRC Press,Boca Raton, FL, 1995.[HR70] E. Hewitt and K. A. Ross.
Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups.Analysis on locally compact Abelian groups . Die Grundlehren der mathematischen Wissenschaften, Band152. Springer-Verlag, New York-Berlin, 1970.[JNR09] M. Junge, M. Neufang, and Z.-J. Ruan. A representation theorem for locally compact quantum groups.
Internat. J. Math. , 20(3):377–400, 2009.[Kah10] B.-J. Kahng. Fourier transform on locally compact quantum groups.
J. Operator Theory , 64(1):69–87,2010.[MVD98] A. Maes and A. Van Daele. Notes on compact quantum groups.
Nieuw Arch. Wisk. (4) , 16(1-2):73–112,1998.[NS06] A. Nica and R. Speicher.
Lectures on the combinatorics of free probability , volume 335 of
London Math-ematical Society Lecture Note Series . Cambridge University Press, Cambridge, 2006. [Obe82] D. M. Oberlin. A convolution property of the Cantor-Lebesgue measure.
Colloq. Math. , 47(1):113–117,1982.[PX03] G. Pisier and Q. Xu. Non-commutative L p -spaces. In Handbook of the geometry of Banach spaces, Vol.2 , pages 1459–1517. North-Holland, Amsterdam, 2003.[RX16] ´E. Ricard and Q. Xu. A noncommutative martingale convexity inequality.
Ann. Probab. , 44(2):867-882(2016).[Rit84] D. L. Ritter. A convolution theorem for probability measures on finite groups.
Illinois J. Math. , 28(3):472–479, 1984.[SS15] A. Skalski and P. M. So ltan. Quantum families of invertible maps and related problems.
Canad. J. Math. ,68(3):698-720, 2016.[So l05] P. M. So ltan. Quantum Bohr compactification.
Illinois J. Math. , 49(4):1245–1270, 2005.[Tak02] M. Takesaki.
Theory of operator algebras. I , volume 124 of
Encyclopaedia of Mathematical Sci-ences . Springer-Verlag, Berlin, 2002. Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5.[VD96] A. Van Daele. Discrete quantum groups.
J. Algebra , 180(2):431–444, 1996.[VD97] A. Van Daele. The Haar measure on finite quantum groups.
Proc. Amer. Math. Soc. , 125(12):3489–3500,1997.[VD07] A. Van Daele. The Fourier transform in quantum group theory. In
New techniques in Hopf algebras andgraded ring theory , pages 187–196. K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2007.[VDN92] D.-V. Voiculescu, K. J. Dykema, and A. Nica.
Free random variables , volume 1 of
CRM MonographSeries . American Mathematical Society, Providence, RI, 1992. A noncommutative probability approachto free products with applications to random matrices, operator algebras and harmonic analysis on freegroups.[Wan95] S. Wang. Free products of compact quantum groups.
Comm. Math. Phys. , 167(3):671–692, 1995.[Wor98] S. L. Woronowicz. Compact quantum groups. In
Sym´etries quantiques (Les Houches, 1995) , pages 845–884. North-Holland, Amsterdam, 1998.
Laboratoire de Math´ematiques, Universit´e de Franche-Comt´e, 25030 Besanc¸on Cedex, France andInstitute of Mathematics, Polish Academy of Sciences, ul. ´Sniadeckich 8, 00-956 Warszawa, Poland
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