Infinitely divisible states on finite quantum groups

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits $n$-th root for all $n\geq1$. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes B\"oge's theorem concerning infinitely divisible measures (commutative case) and Parthasarathy's result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.


Introduction
The space of bounded measures on a compact (semi)group is equipped with a natural convolution operation. The convolution of two probability measures is still a probability measure. Infinitely divisible probability measures are probability measures that admit n-th root for all n ≥ 1, where the root is also a probability measure. On finite groups such probability measures have been shown to be of Poisson type, see [Bög59] and [Sch72].
A positive definite function on a compact group G is a continuous function φ : G → C such that [φ(g −1 i g j )] n i,j=1 is a positive semi-definite matrix for all g 1 , . . . , g n ∈ G and for all n ≥ 1. It is normalized if φ(e) = 1, where e is the unit of G. The pointwise product of two normalized positive definite functions on G is again a normalized positive definite function. From this we can define infinitely divisible normalized positive definite functions on a compact group in a natural way. This is thoroughly studied by Parthasarathy [Par70]. As a special case, he proved that every infinitely divisible normalized positive definite function on a finite group is of Poisson type, although the notion "Poisson type" was not explicitly defined in his paper.
We shall consider the infinite divisibility of states on quantum groups, which provide a more general framework. Our main result is that any infinitely divisible state on a finite quantum group is of Poisson type (in the following also called simply a Poisson state). By taking the finite quantum group to be commutative and cocommutative, we recover the Böge's result [Bög59] of infinitely divisible probability measures and Parthasarathy's result [Par70] on infinitely divisible normalized positive definite functions for finite groups, respectively. We will give two proofs of the main theorem. The first one is based on the ideas of [Sch72] and the second one goes back to [Par70].
The main difficulty in the study of infinite divisibility of states on quantum groups is to capture the "quantum subgroup" on which the states are "supported". It is known that the notion "quantum subgroup" here should be replaced by "quantum hypersubgroup" [FS09b]. Indeed, this is closely related to idempotent states. On a classical compact group, idempotent probability measures are Haar measures on compact subgroups, due to Kawada and Itô [KI40]. In quantum group case, idempotent states not necessarily correspond to quantum subgroups. See [Pal96] for the first counter example on 8-dimensional Kac-Paljutkin quantum group; more discussions can be found in [FS09b]. The right concept one should consider here is the quantum hypergroup [FS09b]. That is first reason why a whole section (Section 2) is devoted to the study of compact quantum hypergroups. The second reason for devoting much effort to compact quantum hypergroups here is the fact that, compared with the theory of compact quantum groups, very little is known for compact quantum hypergroups. The original definition [CV99] is rather technical, which makes it difficult to construct examples. Thus relatively few concrete compact quantum hypergroups are known so far [CV99,Kal01]. This motivates us to present two approaches to constructions of compact quantum hypergroups in Section 2, to enlarge the class of relevant examples. We mention here that although the theory of algebraic quantum groups developed by Delvaux and Van Daele [DVD11a] is very nice, it can not serve as a substitute of compact quantum hypergroups here; see Section 1 and Section 2 for more discussions.
The plan of this paper is as follows. In Section 1 we recall the preliminaries on compact quantum groups, compact quantum hypergroups and algebraic quantum hypergroups. In Section 2 we give two approaches to construct compact quantum hypergroups from compact quantum groups, one induced by an idempotent state and one from a group-like projection. We also give a duality theorem for finite quantum hypergroups. Part of results in this section are new. Section 3 is devoted to the study of Poisson states on compact quantum groups. Finally in Section 4 we prove the main result of this paper, namely that any infinitely divisible state on a finite quantum group is a Poisson state, in two different ways.
1. Preliminaries 1.1. Compact quantum group and its dual. Let us recall some definitions and properties of compact quantum groups. We refer to [Wor98] and [MVD98] for more details.
Any compact quantum group G = (A, ∆) admits a unique Haar state, i.e. a state h on A such that A representation u is called unitary if u is unitary as an element in M n (A), and irreducible if the only matrices T ∈ M n (C) such that uT = T u are multiples of identity matrix. Two representations u, v ∈ M n (A) are said to be equivalent if there exists an invertible matrix T ∈ M n (C) such that T u = vT . Denote by Irr(G) the set of equivalence classes of irreducible unitary representations of G. For each α ∈ Irr(G), denote by u α ∈ A ⊗ B(H α ) a representative of the class α, where H α is the finite dimensional Hilbert space on which u α acts. In the sequel we write n α = dim H α .
Denote Pol(G) = span u α ij : 1 ≤ i, j ≤ n α , α ∈ Irr(G) . This is a dense subalgebra of A. It is well-known that (Pol(G), ∆) is equipped with the Hopf*-algebra structure. That is, there exist a linear antihomormophism S on Pol(G), called the antipode, and a unital *-homomorphism ǫ : Indeed, the antipode and the counit are uniquely determined by 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 ) such that Tr(Q α ) = Tr(Q −1 α ) := d α , which we call quantum dimension of α, and the orthogonal relations hold: The Pontryagin duality can also be extended to compact quantum groups. The dual quantum groupĜ of G = (A, ∆) is defined via its "algebra of functions", which is the C * -algebra defined as the c 0 -direct sum Unless G is finite quantum group,Â is not unital. We defineÂ as the *-algebra via the algebraic direct sumÂ = α∈Irr(G) That is to say, each element ofÂ has only finitely many non-zero components in the direct summands. Clearly,Â is dense inÂ.
We can equipÂ with a discrete quantum group structure. See [VD96] for more details. Recall only that the left Haar weightĥ L onĜ is given bŷ where x α denotes the component of x in the direct summand B(H α ). The right Haar weightĥ R shares a similar form.
On Pol(G) * , the set of bounded linear functionals on Pol(G) (where Pol(G) is viewed with the universal enveloping C * -norm), there is a natural Banach *algebra structure. When ϕ 1 , ϕ 2 are two bounded linear functionals on Pol(G), then their convolution product is defined as ϕ 1 ⋆ ϕ 2 := (ϕ 1 ⊗ ϕ 2 )∆. Clearly we have ϕ 1 ⋆ ϕ 2 ≤ ϕ 1 ϕ 2 , where the norm of functionals on Pol(G) always come from C(G) * . For any ϕ ∈ Pol(G) * , define the involution of ϕ as ϕ * := ϕ(S(·) * ). One can also construct the dual of G = (A, ∆) via the functionals on A with this *-algebra structure, see [MVD98]. Here we use the Fourier transform to say a few words on this.
For a linear functional ϕ on Pol(G), define its Fourier transformφ = (φ(α)) α∈Irr(G) ∈ The Fourier transform F : ϕ →φ sends the convolution to multiplication and is *-preserving: We call G a finite quantum group if the underlying C * -algebra C(G) is finite dimensional. In this case each Q α is identity and h is also a trace, i.e., h(ab) = h(ba) for any a, b ∈ C(G). Then the orthogonal relation becomes where α, β ∈ Irr(G), 1 ≤ i, j ≤ n α , 1 ≤ k, l ≤ n β . If G is a finite quantum group, then so is its dualĜ, and the corresponding Haar weightsĥ L andĥ R coincide, and are denoted byĥ. After normalization the functionalĥ becomes a tracial state of the form:ĥ Moreover, the antipode S satisfies S 2 = ι. Together with * •S • * •S = ι, one obtains directly that S is *-preserving. The Fourier transform F now is a *-isomorphism between the C*-algebras (A, ·, * ) and (A * , ⋆, * ). The notationĥ has some conflict with the Fourier transform of h. It will not be difficult for readers to distinguish them by observing the elements they act on.
1.2. Compact quantum hypergroups and *-algebraic quantum hypergroups. Compact quantum hypergroups were introduced by Chapovsky and Vainerman in [CV99]. Their definition is very technical, relying on the existence of a one-parameter group of automorphisms verifying certain relations. This brings a lot of trouble constructing non-trivial compact quantum hypergroups. Later on Kalyuzhnyi proposed [Kal01] a construction of compact quantum hypergroups using conditional expectations on compact quantum groups. The compact quantum hypergroups discussed in this paper mainly come from this construction. However, we will also give an improvement of Kalyuzhnyi's result in the sense that, a new class of conditional expectations that do not verify Kalyuzhnyi's conditions but still induce compact quantum hypergroups, is constructed. Indeed, such constructions have already been studied by Delvaux and Van Daele on the algebraic level and have also been widely used by others, see for example [FS09b] and [FS09a].
In [DVD11a] Delvaux and Van Daele introduced the so-called *-algebraic quantum hypergroups, which are essentially the algebraic counterparts of compact quantum hypergroups. In a separate note [DVD11b] they gave several constructions and examples of *-algebraic quantum hypergroups. Note that even if it is reasonable to expect that a *-algebraic quantum hypergroup of compact type with positive integrals will yield a compact quantum hypergroup in the sense of [CV99] (see [DVD11a]), it has not been shown so far. We hope that this could be established in the future.
Delvaux and Van Daele's *-algebraic quantum hypergroups admit a very nice biduality theory [DVD11a, Theorem 3.12], that is, for a *-algebraic quantum hypergroup (A, ∆) with its dual (Â,∆), there is a natural isomorphism between (A, ∆) and its bidual (Â,∆). We will however use mainly compact quantum hypergrous in this paper. The reason for this is that, on compact quantum hypergroups a representation theory similar to that of compact quantum groups was developed [CV99]. See Theorem 1.4 below for a Peter-Weyl theory for compact quantum hypergroups.
Let A be a unital C*-algebra equipped with a hypergroup structure as above. An element a ∈ A is called positive definite if ξ · ξ + (a) ≥ 0 for all ξ ∈ A * . It is known that [CV99, Theorem 2.3] if the linear span of all positive definite elements in A is dense in A, then there exists a unique h ∈ A * such that (2) δ is completely positive; (3) (σ t ) t∈R is a continuous one-parameter group of automorphisms of A such that there exists dense subalgebras A 0 ⊂ A and A 0 ⊂ A ⊗ A verifying (a) The one-parameter groups σ t , ι ⊗ σ t and σ t ⊗ ι can be extended to complex one-parameter groups σ z , ι ⊗ σ z and σ z ⊗ ι, z ∈ C, of automorphisms of A 0 and A 0 , respectively; (d) there exists z 0 ∈ C such that h satisfies the strong invariance condition: For short we write H = (A, δ) to denote the compact quantum hypergroup. We can also define the antipode as κ : As one can see, the definition of compact quantum hypergroup is very complicated and technical. Certainly compact quantum groups and the classical compact hypergroups are compact quantum hypergroups [CV99]. But usually it is very difficult to construct other examples. One way to do this uses a sufficiently nice conditional expectation on a compact quantum group, as the following theorem shows.

Then (B, ∆) forms a compact quantum hypergroup.
Like for compact quantum groups, there is a representation theory for compact quantum hypergroups.
for all i, j = 1, . . . , n. Here δ ij denotes the Kronecker symbol. It is called a † - But the orthogonal relation is slightly weaker: We start with a *-algebra A over C with a non-degenerate product. Let M (A) denote its multiplier algebra. A comultiplication, or coproduct on A is a linear *preserving map ∆ : Definition 1.5. [DVD11a] A *-algebraic quantum hypergroup (A, ∆, ǫ, ϕ, S) consists of (1) a *-algebra (A, ∆) with ∆ a comultiplication; (2) a counit ǫ : A → C, which is a *-homomorphism such that (3) a self-adjoint faithful left integral ϕ : A → C: , a ∈ A, which is faithful in the following sense: a ∈ A must be 0 whenever either (4) an antipode S relative to ϕ: We remark here that the left integrals are unique, under the assumption of the existence of ϕ and the antipode S relative to ϕ. Moreover, one can show that the antipode S verifies S(S(x) * ) * = x for all x ∈ A. In the sequel we use simply (A, ∆) for short to denote a *-algebraic quantum hypergroup. Now given a *-algebraic quantum hypergroup (A, ∆), we explain how to construct its dual (Â,∆), which is again a *-algebraic quantum hypergroup.
By taking the dual of (Â,∆) again, we recover (A, ∆). This is the biduality for *algebraic quantum hypergroups [DVD11a, Theorem 3.12]. Moreover, a *-algebraic quantum hypergroup is of compact type if and only if its dual is of discrete type. Here, a *-algebraic quantum hypergroup (A, ∆) is said to be of compact type if A possess an identity 1 (and thus ∆(1) = 1 ⊗ 1). And a *-algebraic quantum hypergroup (A, ∆) is of discrete type if it is equipped with a co-integral h ∈ A, that is a non-zero element such that ah = ǫ(a)h for all a ∈ A.
We close this subsection with a question proposed by Delvaux and Van Daele at the end of [DVD11a]. Certainly a compact quantum hypergroup is a *-algebraic quantum hypergroup is of compact type. But will a *-algebraic quantum hypergroup of compact type with positive integrals always yield a compact quantum hypergroup?
1.2.3. Finite quantum hypergroups. Our main result lies in the framework of finite quantum hypergroups, which are compact quantum hypergroups whose underlying C*-algebras are finite dimensional. The reader should be careful that this notion is different from the one in [FS09b], where one discusses a similar concept introduced only on the algebraic level.
In a finite quantum hypergroup H = (A, ∆), the set of matrix elements of all inequivalent irreducible † -representations {u α ij : 1 ≤ i, j ≤ n α , α ∈ Irr(H)} form a basis of A. The underlying C*-algebraÂ of its dual (as a *-algebraic quantum hypergroup) (Â,∆), is nothing but A * . Let ω α ij be the dual basis of u α ij in A * , then from Hence {ω α ij : 1 ≤ i, j ≤ n α , α ∈ Irr(H)} can be viewed as the matrix units of the dual of H.

Examples of compact quantum hypergroups
In this section we present some new constructions of compact quantum hypergroups, which will be of use in the later sections of the paper.
2.1. Examples from idempotent states. Suppose that G = (A, ∆) is a compact quantum group. Denote by A * the set of all bounded linear functionals on A. For any ϕ 1 , ϕ 2 ∈ A * , we can define their convolution product, as we did on Pol(G): We ignore ⋆ if no ambiguity occurs.
Denote by S(A) the set of states on We use Idem (G) to denote the set of idempotent states on G. Observe first that if φ ∈ Idem(G), we have φ = φS on Pol(G) [FS09b]. In other words,φ(α) is a projection for each α ∈ Irr(G); as it is contractive, it is also self-adjoint.
Now we turn to the study of coidalgebras. Most of the results in this subsection can be found in [DFW17] and [FS09a]. A left (resp. right) coidealgebra C in a compact quantum group The first lemma is a special case of [FS09a, Lemma 3.1], and also a variation of [MVD98, Lemma 4.3].
The next lemma lists some useful properties of these maps. Proof.
(1)-(4) are just straightforward computations. For (5) we prove the statement only for E ℓ φ , as the proof for E r φ is similar. For this note that it suffices to show the first equation for any a, b the coefficients of unitary representation of G. The case for general a, b follows from the density argument. Let u α ij , 1 ≤ i, j ≤ n α be the coefficients of the irreducible representation u α , α ∈ Irr (G). Since φ ⋆ φ = φ, we have So for any u α ij and u β kl we have Now apply (2.2) for b = u β tq and use (2.1), we get The remaining part is a consequence of the equality E ℓ φ (1) = 1.
The map E φ shares the similar properties of E ℓ φ and E r φ , see the following lemma (1)-(4). Moreover, E φ commutes with the antipode S, as the following lemma (5) shows, which means that the algebra A φ := E φ (A) possesses nicer properties than E ℓ φ (A) and E r φ (A). That is why we use the map E φ in the remaining part of the paper.
Hence A φ is a unital C * -subalgebra of A. Moreover, Proof. Again we omit the proof of (1)-(4) here. The fact that A φ is a unital C *subalgebra follows directly from these properties. To prove (5), it suffices to check the equality for u α ij , the coefficients of unitary representation of G, where 1 ≤ i, j ≤ n α , α ∈ Irr(G). And that is a consequence of where the last equality follows from the facts that φ = φS on Pol(G) and S(u α lj ) = (u α jl ) * .
The following proposition says that E φ as above is a projection verifying all the conditions in Theorem 1.3. Thus (A φ , ∆ φ ) becomes a compact quantum hypergroup.
Proposition 2.4. Let G = (A, ∆, ǫ, S) be a compact quantum group. Let E φ be defined as above and set (3) the restriction of ǫ to A φ is a counit, i.e., Hence by Theorem 1.3, (A φ , ∆ φ ) is a compact quantum hypergroup.
Proof. The proof is now based on straightforward computations using the last two lemmas.
Let φ be an idempotent state on compact quantum group G = (A, ∆). A func- In the remaining part of this subsection we characterize the φ-bi-invariant functionals, where φ is an idempotent state. It turns out that one can transfer each φ-bi-invariant functional on A to its restriction to A φ , preserving the norm and the *-algebra structure. See also [DFW17] for related work.
We formulate the results of φ-bi-invariant functionals here without the proof.
Lemma 2.5. Let φ ∈ Idem(G) and u ∈ A * . Then u is φ-bi-invariant if and only if u = u| A φ E φ . In this case, the following hold: (2) u is positive (resp. a state) if and only if u| A φ is positive (resp. a state); Remark 2.6. The last property implies that the idempotent state φ can be recovered by the counit ǫ through this formula. This will be frequently used in the sequel.

Examples from group-like projections.
Group-like projections in algebraic quantum groups were first introduced by Van Daele and Landstad in [LVD08]. The relation between idempotent states and group-like projections has been studied by Franz and Skalski in [FS09b] on compact quantum groups, and then by Faal and Kasprzak in [FK17], and Kasparzak and So ltan in [KS18] on locally compact quantum groups.
The main result in this subsection is that a group-like projection in a compact quantum group induces a compact quantum hypergroup. The main ingredients were obtained on the algebraic quantum group level by Delvaux and Van Daele [DVD11b]. Note that by taking adjoints we have The following proposition is not a direct consequence of Theorem 1.3, since the projection P is not h-invariant in general. But one can check the proof [Kal01, Theorem 2.1] to see that this is only used to deduce the strong invariance (1.2) of h, which is nothing but (1) of the Proposition 2.8.
Proposition 2.8. Let (A, ∆) be a compact quantum group. Let p be a group-like projection in A, then P : A → A p := pAp, a → pap is an h-preserving conditional expectation such that (2) (P ⊗ P )∆ = (P ⊗ P )∆P ; (3) SP = P S; (4) the restriction of ǫ to A p is a counit, i.e., where ∆ p := (P ⊗ P )∆| Ap . Then (A p , ∆ p ) is a compact quantum hypergroup.

Proof.
(1) For any a, b ∈ A, we have by (2.3) (2) For any a ∈ A, we have by (2.3) and (2.4) (3) For any a ∈ A, it follows from the definition of group-like projection that

SP (a) = S(pap) = S(p)S(a)S(p) = pS(a)p = P S(a).
(4) By definitions of the counit ǫ on A and the group-like projection p we obtain where we have used un easy fact ǫ(p) = 1 in the last equality. So (ǫ⊗ι)∆ p = ι and the proof of the other equality is similar.
2.3. A duality theorem. Let (A, ∆) be a finite quantum group. Let φ be an idempotent state on A. Denote by (A φ , ∆ φ ) the finite quantum hypergroup induced by φ. Considered as an element inÂ, p = φ is a group-like projection [FS09b]. We then let (Â p ,∆ p ) be the finite quantum hypergroup associated to the group-like projection p = φ inÂ. We show that (Â p ,∆ p ) is the dual of (A φ , ∆ φ ).

Poisson states on compact quantum groups
Let G = (A, ∆) be a compact quantum group. For each φ ∈ Idem(G), we say that {ω t } t≥0 is a convolution semigroup of functionals on A starting from φ if If moreover, each ω t ∈ S(A), we call {ω t } t≥0 a convolution semigroup of states starting from φ. We say that the convolution semigroup of states Then it is easy to check that {exp φ (tu)} t≥0 form a norm continuous convolution semigroup of functionals. We aim to find sufficient and necessary conditions on u such that {exp φ (tu)} t≥0 is a convolution semigroup of states. For this we make some notations. A functional u ∈ A * is called Hermitian if u (x * ) = u (x) for all x; it is further called conditionally positive definite with respect to φ if u (x * x) ≥ 0 for all x such that φ (x * x) = 0. The main theorem in this section is as follows.
Theorem 3.1. Suppose that G = (A, ∆) is a compact quantum group. Let φ ∈ Idem (G). Then for u ∈ A * , the following statements are equivalent.
The following proposition proves Theorem 3.1 on general unital C * -algebras, under the additional assumptiion that φ = ε is a character (the quantum group structure can be then removed, since any u ∈ A * is ǫ-bi-invariant with ǫ the counit).
Proposition 3.2. Let A be a unital C * -algebra with ε a character. Then for any non-zero bounded linear functional u on A such that u(1) = 0 and u(x * x) ≥ 0 for all ε(x * x) = 0, we have u = r(v − ε), where r > 0 and v is a state.
Proof. Note first that ε(x * x) = |ε(x)| 2 . So ε(x * x) = 0 if and only if x ∈ ker ε = {x : ε(x) = 0}. Let u 0 := u| ker ε be the restriction of u to ker ε. By assumption, u 0 is a bounded linear positive functional on the ideal ker ε. So it admits a unique positive linear extension u 0 to A such that u 0 | ker ε = u 0 and u 0 = u 0 . Hence for any x ∈ A, we have x − ε(x)1 ∈ ker ε and thus where r := u 0 = u 0 > 0 and v := 1 r u 0 is a state.
Now we are ready to prove Theorem 3.1. The idea is to restrict the problem to A φ , and then apply Proposition 3.2 by recovering the idempotent state φ as Proof of the Theorem 3.1. The direction (1) ⇒ (2) is clear. To prove (2) ⇒ (1), suppose u = 0 and write u = u| A φ E φ by Lemma 2.5. Note that ǫ| A φ is a character on the unital C * -algebra A φ . From the definition of u, we have u| A φ (1) = 0. Moreover, for any x ∈ A such that ǫ| A φ E φ (x) * E φ (x) = 0, we have by Lemma 2.2 and Lemma 2.5 that Again, by Lemma 2.2 and Lemma 2.5, the conditionally positive definiteness of u with respect to φ implies

So we have by Proposition 3.2 that u|
as desired.
Definition 3.3. Let φ ∈ Idem(G). We denote by N φ (G) the class of u ∈ A * that satisfy the conditions in Theorem 3.1. By condition (2), ω = exp φ (u) is a state for each u ∈ N φ (G). Denote by P φ (G) the set of all such states. Set P(G) := φ∈Idem(G) P φ (G). Then any ω ∈ P(G) is said to be of Poisson type, or a Poisson state on G.
Recall that any norm continuous convolution semigroup of states {ω t } t≥0 on a compact quantum group G = (A, ∆) can be recovered by exponentiation with the bounded generator u := lim t→0 + 1 t (ω t − ω 0 ). It is not difficult to see that u(1) = 0 and u is conditionally positive definite with respect to ω 0 , since these hold for each 1 t (ω t − ω 0 ), t > 0. Then together with Theorem 3.1 we have the following result. Theorem 3.4. Let φ be an idempotent state on a compact quantum group G = (A, ∆). For any non-zero bounded linear functional ω on A such that ωφ = φω = ω, the following are equivalent (1) ω = ω 1 with {ω t } t≥0 a norm continuous convolution semigroup of states such that ω 0 = φ;

Infinitely divisible states on finite quantum groups
In this section we prove the main result of the paper.
Definition 4.1. Let G = (A, ∆) be a compact quantum group. A state ω ∈ S(A) is said to be infinitely divisible if ω = ω n n for some ω n ∈ S(A) for all n ≥ 1. We use I(G) to denote the set of all infinitely divisible states on G.
Clearly Poisson states are infinitely divisible. Our main result in this section is that any infinitely divisible state on a finite quantum group is a Poisson state. From now on, unless stated otherwise, G = (A, ∆) always denotes a finite quantum group.
The following lemma is well-known, and the proof follows from standard arguments. ii . As a direct consequence, we have the following Jordan type decomposition, which is quite easy but very helpful.  Proof. The first part is a consequence of the Lemma 4.2, the discussion at the end of section 1 on finite quantum hypergroups, and the biduality of quantum hypergroups. To show (4.1), assume α 0 ∈ Irr(Ĥ) corresponds to the trivial representation, i.e., v α0 = 1. Then Theorem 1.4 and the positive semi-definiteness of [a α ij ] yield: Recall that if G, is finite, A is finite dimensional, so A = Pol(G) andÂ =Â. The Fourier transform F is an isomorphism between Banach *-algebra (A * , · ) and finite-dimensional C*-algebra (Â, · ).
Let φ be an idempotent state on finite quantum group G = (A, ∆). For any u ∈ A * such that u = uφ = φu and u − φ < 1, define the logarithm of u with respect to φ as Then we have the following properties of logarithm and exponential.
, if uv = vu and the following holds: (5) If moreover, u is a state such that u − φ < 1 2 and u n − φ < 1 2 for some n ≥ 1, then Consequently, in such a case we have Proof.
Then the proof for n + 1 is finished, which shows (5).
As this proposition suggests, to show that an infinitely divisible state is of Poisson type, it is important to capture the corresponding idempotent state. For this we need two lemmas. The first one is an easy fact in matrix theory. Proof. Since P is a self-adjoint projection, we may assume without loss of generality that where I r is the identity in M r (C) with r = rank(P ). From A = AP = P A and AB = P it follows with A r B r = I r . Note that So A r = B r = 1. This is to say, Then all the eigenvalues of A * r A r must be 1 and thus A * r A r = I r . Hence B r = A * r and thus A * A = AA * = P . To show the remaining part, note that the Fourier transforms of u, v are in some full matrix algebra M n (C). Since F is a contraction, we have From the fact that the Fourier transform F is a *-homomorphism it followŝ u =ûφ =φû,ûv =φ.
Becauseφ is a self-adjoint projection, the previous argument yieldsû * û =ûû * =φ. Again, since F is a *-homomorphism, u * u = uu * =φ. Thus u * u = uu * = φ, by the injectivity of F . Lemma 4.9. Let H = (A, ∆) be a finite quantum hypergroup with the dualĤ = (Â,∆), which is also a finite quantum hypergroup. Suppose that u is a state on A such that uu * = u * u = ǫ, where ǫ is the counit of A. Then u is an n-th root of ǫ for some n ≤ dimÂ. If, moreover, H is a finite quantum group, then u is also a character.
If moreover, H is a finite quantum group, then we can obtain a slightly stronger estimate. Indeed, by choosing {v α ij } to be unitary irreducible representations, we have from the orthogonal relation (1.1) that where the first inequality holds because [a α ij ] nα i,j=1 is positive semi-definite with trace 1. Recall that p α ≥ 0 and α∈Irr(Ĥ) p α = 1, thus This happens only if p α ′ = 1 and n α ′ = 1 for some α ′ ∈ Irr(Ĥ). That is to say, u = v α ′ is a one dimensional unitary representation ofĤ, thus a character.
The following proposition, gathering the main ingredients of preceding lemmas, will be used to prove Theorem 4.11. Proposition 4.10. Let G = (A, ∆) be a finite quantum group with the counit ǫ. Suppose that u, v ∈ S(A) and φ ∈ Idem(G) are such that u = uφ = φu and uv = φ. Then there exists a positive integer m ≤ dimÂ such that u m = φ.
Proof. From Lemma 4.8 it follows u * u = uu * = φ. Let u 0 and ǫ 0 be the restrictions of u and ǫ to A φ , respectively. Then u 0 is a state on finite quantum hypergroup H := (A φ , ∆ φ ), and ǫ 0 is the counit on H such that u 0 u * 0 = ǫ 0 . Note that by Theorem 2.3, the dual of H is (Â p ,∆ p ), which is again a finite quantum hypergroup, where p = φ is considered as a group-like projection inÂ. So Lemma 4.9 implies  The first proof. P(G) ⊂ I(G) is clear. Let ω ∈ I(G). We claim that for any positive integer N ≥ 2, there exists a sequence {b n } n≥0 of roots of ω such that b 0 = ω, b n−1 = b N n , n ≥ 1. Indeed, since A is finite dimensional, the set of states Z = S(A) is compact with respect to the norm topology. Thus j≥0 Z j , where Z j = Z for all j, is compact with respect to the product topology. Let a n ∈ Z be any n-th root of ω for all n ≥ 0. Then the sequence of non-empty closed sets is decreasing: W 1 ⊃ W 2 ⊃ · · · , and thus any finite intersection of {W k } k≥1 is non-empty. By compactness of j≥0 Z j , k≥1 W k = ∅. Hence one can choose (b 0 , b 1 , . . . ) ∈ k≥1 W k , which verifies b 0 = ω, b n−1 = b N n , n ≥ 1. This proves the claim.
Choose N = (dimÂ)! ≥ 2 and let {b n } n≥0 be as above. Since Z is compact, there exists a subsequence {c j } j≥0 of {b i } i≥0 such that c j converges to some c ∈ Z. If we fix a non-negative integer i, we have b i = c rj j for sufficient large j and some integer r j ≥ N ≥ 2. That is, We can assume that c rj−1 j converges to some d i ∈ Z, otherwise consider some This implies b i ∈ cZ ∩ Zc for all i ≥ 0. From the choice of c j we have c j ∈ cZ ∩ Zc for all j ≥ 0. Then for any i the corresponding c rj −1 j ∈ cZ ∩ Zc for all j, which implies that d i ∈ cZ ∩ Zc by the compactness of cZ ∩ Zc. Now considering (4.3) for {c j } j≥0 , instead of {b i } i≥0 , we obtain an updated version of (4.4): (4.5) c j = cd ′ j = d ′ j c, j ≥ 0, where d ′ j ∈ cZ ∩ Zc. Letting j → ∞, consider the subsequence of {d ′ j } j≥0 if necessary, one obtains (4.6) c = cd = dc, where d ∈ cZ ∩ Zc by the compactness of cZ ∩ Zc. Suppose d = ce for some e ∈ Z, then d 2 = dce = ce = d, i.e., d is an idempotent state. By Proposition 4.10, we obtain c m = d for some m ≤ dimÂ. Then by choosing N to be (dimÂ)!, we have Denote by φ the idempotent state d. Set ω 0 := ω and ω n := c N n for all n ≥ 1. Then ω n → φ as n tends to ∞. By definition, {ω n } n≥0 is a subsequence of {b j } j≥0 , thus ω n−1 = ω sn n with s n = N r n for all n ≥ 1. Moreover, from (4.5) we have ω n = c N n = (cd ′ n ) N = c N d ′ N n = φd ′ N n = φ(φd ′ N n ) = φω n , n ≥ 0. Similarly, ω n = ω n φ, n ≥ 0. Hence {ω n } n≥0 verifies the conditions of Proposition 4.7, and consequently ω ∈ P φ (G).
which tends to 0 as k → ∞. This shows ω = exp φ (u) and finishes the proof.
The second proof of Theorem 4.11. Again, P(G) ⊂ I(G) is clear. Let ω ∈ I(G). From the first proof we know that there exist an idempotent state φ ∈ Idem(G) and a sequence of roots {ω n k } k≥0 ⊂ S(A) with {n k } k≥0 an increasing sequence of positive integers such that ω n k n k = ω, ω n k = ω n k φ = φω n k , k ≥ 0, and ω n k → φ as k → ∞. Let u and u n k be the restrictions of ω and ω n k to A φ for all k ≥ 0, respectively. Then from Lemma 2.5 u is a state on finite quantum hypergroup H = (A φ , ∆ φ ) such that {u n k } k≥0 is a sequence of roots of u in S(A φ ) verifying u n k n k = u and u n k → ǫ 0 , k → ∞, where ǫ 0 is the counit of H. Now we repeat a calculation in Lemma 4.9. Let Irr(Ĥ) be the set of unitary equivalent classes of irreducible unitary representations ofĤ.
Let v α ij be the matrix elements corresponding to the representation. Then we can write u n k as u n k = α∈Irr(Ĥ) p α,k nα i,j=1 a α,k ij v α ij with p α,k ≥ 0, α∈Irr(Ĥ) p α,k = 1 and [a α ij ] nα i,j=1 and [a α,k ij ] nα i,j=1 positive semi-definite with trace 1 for each k.
So p α0,k → 1 as k → ∞. Let m := rank(ǫ 0 ). Then A φ can be viewed as a subalgebra of M m (C). We use · p to denote the Schatten p-norm of matrices. By Hölder's inequality, (4.9) û 2/n k = û n k n k 2/n k ≤ û n k n k 2 , k ≥ 0. Let λ 1 , . . . , λ m be all singular values ofû. Then all λ i are non-zero. To see this, it suffices to show thatû is invertible. Note thatǫ 0 is the identity matrix in M m (C). Since for large k there holds û n k −ǫ 0 ≤ u n k − ǫ 0 < 1, we have thatû n k is invertible for large k, and so isû.
Combining this with (4.8), we have for all k ≥ K, where M is a constant independent of k. Here we have used the fact that λ i > 0 for all i. From Lemma 2.5 it follows that sup k≥1 n k ω n k − φ = sup k≥1 n k u n k − ǫ 0 < ∞.
Remark 4.13. Both proofs rely on the capture of idempotent state where the infinitely divisible state is "supported on" and the sequence of roots converging to this idempotent state. After this the first proof aims to show that this sequence of roots can chosen to form a submonogeneous convolution semigroup (Proposition 4.7 (3)), while the idea of the second proof is derived from a general result Proposition 4.12, concerning the decay property (4.10) of this sequence of roots. The inequality (4.10) also allows us to simplify the proof of the main theorem in [Par70] for the finite case.