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≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 1$$\end{document}. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Böge’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 [2] and [18].
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 [17]. 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 or cocommutative, we recover the Böge's result [2] of infinitely divisible probability measures and Parthasarathy's result [17] 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 [18] and the second one goes back to [17].
The plan of this paper is as follows. In Sect. 2 we recall the preliminaries on compact quantum groups and introduce the notion of Plancherel triples. In Sect. 3 we explain how to construct a Plancherel triple from an idempotent state on a finite quantum group. Section 4 is devoted to the study of Poisson states on compact quantum groups. Finally in Sect. 5 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.

Compact quantum group and its dual
Let us recall some definitions and properties of compact quantum groups. We refer to [21] and [15] for more details. Here and in the following, ι always denotes the identity map. We denote G = (A, Δ) and A = C(G). For simplicity, we write Δ (2) = (Δ ⊗ ι)Δ.
Any compact quantum group G = (A, Δ) admits a unique Haar state, i.e. a state h on A such that Consider an element u ∈ A ⊗ B(H ) with dim H = n. By identifying 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 i j : 1 ≤ i, j ≤ n α , α ∈ Irr(G) . This is a dense subalgebra of A. On Pol(G) the Haar state h is faithful. It is well-known that (Pol(G), Δ) 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 : Pol(G) → C, called the counit, such that Here m denotes the multiplication map m : Indeed, the antipode and the counit are uniquely determined by Now we add a remark on the C*-norms on Pol(G). We are interested in the following two C*-norms on Pol(G): 1. the univesal norm: 2. the reduced norm: where π h is the GNS representation associated with the Haar state h.
We shall denote by C u (G) and C r (G) the completions of Pol(G) with respect to · u and · r , respectively. Then the comultiplication Δ and the Haar state h on Pol(G) admit extensions to C u (G) (resp. C r (G)), denoted by Δ u and h u (resp. Δ r and h r ), respectively. Both (C u (G), Δ u ) and (C r (G), Δ r ) form compact quntum groups.
Note that the counit can be always extended to C u (G). While, this is not always the case for C r (G). If can be also extended to C r (G), then G is said to be coamenable. An equivalent definition is, G is coamenable iff · r = · u . Note that · r ≤ · u always holds. We refer to [1] for more information. Throughout this paper, we shall always consider compact quantum group G on the universal level, so that the counit can always be extended to the C(G).
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: where α, β ∈ Irr(G), 1 ≤ i, j ≤ n α , 1 ≤ k, l ≤ n β . We call G a finite quantum group if the underlying C * -algebra C(G) is finite dimensional. Note that when G is finite, we have C(G) = Pol(G) and then G is coamenable. In this case each Q α is identity and h is 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 . Moreover, the antipode S satisfies S 2 = ι. Together with * • S • * • S = ι, one obtains directly that S is *-preserving. The Pontryagin duality can also be extended to compact quantum groups. We only explain here for finite quantum groups. If G = (A, Δ) is a finite quantum group, then we may construct its dualĜ = (Â,Δ) as follows. The underlying finite dimensional C*-algebraÂ ofĜ is defined as A , the set of all (bounded) linear functionals on A. For ϕ 1 , ϕ 2 ∈Â, their convolution product is defined as ϕ 1 ϕ 2 := (ϕ 1 ⊗ ϕ 2 )Δ. We may define the involution onÂ as: ϕ * := ϕ(S(·) * ). Then (Â, , * ) becomes a finite dimensional C*-algebra. For each ϕ ∈Â, and it is easily seen thatΔ defines a comultiplication onÂ. HenceĜ = (Â,Δ) becomes a finite quantum group. Moreover, it is equipped with a Hopf*-algebra structure with the antipodeŜ and the counit given byŜ(ϕ) := ϕ S andˆ (ϕ) := ϕ(1 A ), respectively. Starting from the finite quantum groupĜ = (Â,Δ), we may also construct its dual. Then in this way we recover the quantum group G = (A, Δ).
The Fourier transform F on a finite quantum group where h is the Haar state on G. We shall useâ to denote the its Fourier transform F (a) for simplicity. Then we have the following Parseval's identity:ĥ whereĥ is the Haar state onĜ and c > 0 is a constant. If we consider the Fourier transform fromÂ to A, then we obtain a similar equation to (2.4).

Plancherel triple
In this subsection we introduce the notion of Plancherel triple, which is slightly different from the same notion in [20]. Moreover, one can compare it with the so-called D-pairs in [13]. Let A, B be two finite dimensional C*-algebras. Suppose that  where p A ∈ A and p B ∈ B are support projections of A and B , respectively, i.e., 4. The Fourier transforms F A : A → B, a →â and F B : B → A, b →b satisfy: for all a 1 , a 2 ∈ A, b 1 , b 2 ∈ B and some c, c > 0, whereâ andb are defined through: Note that the above definition is self-dual, that is, We have the following properties of a Plancherel triple.

Proposition 2.3 Let (A, B, , ) be a Plancherel triple. Suppose that
for all α, β, i, j, k, l, where c is the constant appearing in (2.5).

Plancherel triple induced from an idempotent state
In this section we will construct a Plancherel triple from an idempotent state on a finite quantum group G = (A, Δ). An idempotent state on G is a state φ such that φ φ = φ. It is well-known that (see for example [9]) considered as an element inÂ, p = φ is a group-like projection inÂ. By a group-like projection of the finite quantum groupĜ = (Â,Δ) we mean a non-zero element p ∈Â such that p = p * = p 2 and is a C*-subalgebra of A,Â p := pÂ p is a C*-subalgebra ofÂ and the bilinear form is inherited from the one on the pair (A,Â). In the following we shall explain the constructions of A φ andÂ p in detail. Before this we remark here that these constructions have already been studied by many people [5,6,8,9]. Many results are well-known and their proofs are omitted here.

C*-subalgebra A
In this subsection we construct and study the C*-subalgebra A φ . Recall that an idempotent state on a compact quantum group G is a state φ such that φ φ = φ. Denote by Idem(G) the set of all idempotent states on G. Recall also that if φ ∈ Idem(G), we have φ = φ S on Pol(G), where S is the antipode on Pol(G) [9]. We use A to denote the set of all bounded linear functionals on A.
The first lemma is a special case of [8, Lemma 3.1], and also a variation of [15,Lemma 4.3].

Lemma 3.1 Let φ be an idempotent state on a compact quantum group
On the other hand, The map E φ shares the similar properties of E 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. This enables the algebra E φ (A) to possess nicer properties 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 where the last equality follows from the facts that φ = φ S on Pol(G) and S(u α l j ) = (u α jl ) * .

-bi-invariant functionals
Let φ be an idempotent state on a compact quantum group In this subsection we characterize the φ-bi-invariant functionals. 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 [4] for related work.
We formulate the results of φ-bi-invariant functionals here without the proof. Note that in the sequel we shall use u to denote the norm of u ∈ A as a functional on A.
In this case, the following hold: (2) u is a positive linear functional (resp. a state) on A if and only if u| A φ is a positive linear functional (resp. a state) on A φ ;

The Plancherel triple (A ,Â p , , )
Let G = (A, Δ) be a finite quantum group. Let φ be an idempotent state on G, then p = φ is a group-like projection inÂ. By Lemma 3.3, A φ = E φ (A) is a finite dimensional C*-algebra. ClearlyÂ p = pÂ p is also a finite dimensional C*-algebra. Moreover, Lemma 3.4 implies that (A φ ) =Â p . The main result of this section is the following proposition. 1. In this case, we claim that the comultiplications on A φ andÂ p are respectively . Then automatically they are positive. Indeed, by definition, Since , is non-degenerate, we have Δ A φ = Δ φ . This proves the claim. Similarly, one can show ΔÂ p =Δ p . 2. The counits on A φ andÂ p are respectively φ := | A φ andˆ p :=ˆ |Â p . Here andˆ are respectively the counits on A andÂ. In fact, by definition,

H. Zhang
Then it is easy to check that φ : A φ → C andˆ p :Â p → C are both *-isomorphisms. 3. On one hand, the support projection pÂ ofˆ = Â verifies Therefore the support projection ofˆ p is pÂ p = pÂ. Thus the Haar functional on A φ is On the other hand, by Lemma 3.3, That is, p A φ = q is the support projection of φ . So the Haar functional onÂ p iŝ Hence the Haar functionals h φ andĥ p are faithful, positive and tracial.

For any a ∈
Indeed, by (2.1), Since φ = φ S and E φ (a) = a, we have Hence φ h(·a) = h(·a), for all a ∈ A φ . From (2.2) and E r φ (a) = a one can deduce in a similar way that h(·a) φ = h(·a).
This proves our claim. Now for any a ∈ A φ we have

Hence the Parsevel's identity (2.4) on the pair (A,Â) yieldŝ
Similarly, one can show that for some constant c > 0.

Let G = (A, Δ) be a compact quantum group. Denote by S (A) the set of all states on A.
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 Recall here that u denotes the norm of u ∈ A as a functional on A. Moreover, it is a Banach norm, since 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 4.1 Suppose that G = (A, Δ) is a compact quantum group. Let φ ∈ Idem (G).
Then for u ∈ A , the following statements are equivalent.
The following proposition proves Theorem 4.1 in the level of unital C * -algebras. It can be considered as a special case of Theorem 4.1 when φ = is the counit. In this case each u ∈ A is -bi-invariant and φ = is a character.

Proposition 4.2 Let A be a unital C * -algebra with a character. Then for any non-zero bounded linear functional u on A such that u(1 A
So (x * x) = 0 if and only if x ∈ ker = {x : (x) = 0}. Let u 0 := u| ker be the restriction of u to ker . By assumption, u 0 is a bounded linear positive functional on the ideal ker . So it admits a unique positive linear extension u 0 to A such that u 0 | ker = u 0 and u 0 = u 0 . Hence for any x ∈ A, we have x − (x)1 A ∈ ker 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 4.1. The idea is to restrict the problem to A φ , and then transfer the decomposition from A φ to A.

Proof of Theorem 4.1 The direction (2) ⇒ (1) is clear. To prove (1) ⇒ (2)
, suppose u = 0 and write u = u| A φ E φ by Lemma 3.4. Note that | A φ is a character on the unital C * -algebra Again, by Lemma 3.2 and Lemma 3.4, the conditionally positive definiteness of u with respect to φ implies

So we have by Proposition 4.2 that u|
as desired. Recall that any norm continuous convolution semigroup of states {ω t } t≥0 on a compact quantum group G = (A, Δ) can be recovered by exponentiation with a bounded generator u := lim t→0 + 1 t (ω t − ω 0 ). It is not difficult to see that u(1 A ) = 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 4.1 we have the following result.

Theorem 4.4 Let φ be an idempotent state on a compact quantum group G = (A, Δ).
For any non-zero bounded linear functional ω on A such that ωφ = φω = ω, the following are equivalent

Remark 4.5
The convolution semigroup of states {ω t } t≥0 on a compact quantum group G is said to be weakly continuous if ω t (a) → ω 0 (a), t → 0 + for any a ∈ C(G). Clearly norm continuous convolution semigroup of states is weakly continuous. When G is a finite quantum group, the converse also holds. But for general compact quantum group G, there exists weakly continuous convolution semigroup of states which is not norm continuous. In this case, the generator is unbounded.

Infinitely divisible states on finite quantum groups
In this section we prove the main result of the paper. G = (A, Δ) 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) and for all n ≥ 1. We use I (G) to denote the set of all infinitely divisible states on G.

Definition 5.1 Let
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, Δ) always denotes a finite quantum group.
The following lemma is well-known, and the proof follows from standard arguments. As a direct consequence, we have the following decomposition, which is quite easy but very helpful.

Corollary 5.3 Let
Let G = (A, Δ) be a finite quantum group. Then for any u ∈ A =Â we denote by u Â the C*-norm of u as an element in C*-algebraÂ. Recall that u is the norm of u as a functional. Moreover, we have u Â ≤ u .
Let φ be an idempotent state on finite quantum group G = (A, Δ). 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. G = (A, Δ) is a finite quantum group. Let φ be an idempotent state on A, then for any bounded linear functionals u, v on A such that u = uφ = φu and v = vφ = φv, we have

Lemma 5.5 Suppose that
, 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 (1)-(4) are direct and hold for all Banach algebras. To show (5), let u φ be the restriction of u to A φ . Then by Lemma 3.4, u φ is a state on a finite-dimensional C * -algebra A φ . Moreover, i j ω k i j verifies the condition of Corollary 5.3 and it follows that Now we show (5) by the induction argument. Clearly it holds for n = 1. Suppose for now that it holds for n. Set r := v 1 , s := v n and t : it follows that v n+1 = v 1 + v n + v 1 v n and thus by Lemma 3.4 (4) w n+1 = w 1 + w n + w 1 w n . This, together with (5.1) and Lemma 3.4 (1), yields Then By assumption, 1 − 2r , 1 − 2t > 0, so 1 − 2s > 0. Hence u and u n verify the conditions in (4), and we obtain where in the second equality we have used the induction for n. This finishes the proof for n + 1 and then shows (5).

Remark 5.6
In fact, to prove (5) we have used the fact that u Then ω ∈ P φ (G).
Then by the definition of ω j 0 and Lemma 5.5 (1) (2), To prove ω ∈ P φ (G), it suffices to show that v ∈ N φ (G). For this we check that v verifies Theorem 4.1 (1). Clearly, v(1 A ) = 0, since ω j 0 is a state. By the definition of logarithm, v = vφ = φv. It remains to show that for any where N j := n j 0 · · · n j−1 . Recall that for all j ≥ j 0 , ω m j − φ < 1/2. Thus by Lemma 5.5 (5) we have and by Lemma 5.5 (1) The condition (1) implies that N j → ∞ as j → ∞. Now for any x ∈ A such that φ(x * x) = 0, we have Letting j → ∞, we have v 0 (x * x) ≥ 0, which ends the proof.
As this proposition suggests, to show that an infinitely divisible state is of Poisson type, it is important to capture the idempotent state where the infinitely divisible state is "supported on". 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 The remaining part follows from the facts that p = φ is a self-adjoint projection inÂ and (A, B, , )

Lemma 5.9 Let
Since v and v * are both positive functionals, vv * (·) = Δ A (·), v ⊗ v * is also a positve functional on A. Hence we have by Corollary 5.4 that So we have That is to say, either h B (u) = 0 or u = A . Since for any n ≥ 1, u n is again a state such that u n u * n = A , we obtain, by a similar argument, that either h B (u n ) = 0 or u n = A . If u is not a n-th root of A for all 1 ≤ n ≤ dim B, then we have Thus a 0 = 0, which leads to a contradiction. So we must have u m = A for some 1 ≤ m ≤ dim B.
The following proposition, gathering the main ingredients of preceding lemmas, will be used to prove Theorem 5.11. Proof From Lemma 5.8 it follows u * u = uu * = φ. Let u φ and φ be the restrictions of u and the counit of A to A φ , respectively. From Proposition 3.5 (A φ ,Â p , , ) forms a Plancherel triple. By Lemma 3.4, u φ is a state on A φ such that u φ u * φ = φ . So Lemma 5.9 implies u m φ = φ for some m ≤ dim A φ ≤ dimÂ. Hence Lemma 3.4 gives u m = u m Now we are ready to prove the main result of this paper.
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 such that 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 nonnegative integer i, we have b i = c r j j for sufficient large j and some integer r j ≥ N ≥ 2. 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 s n n with N |s n for all n ≥ 1. Moreover, from (5.5) 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 5.7, and consequently ω ∈ P φ (G). for all k ≥ K . Since u is invertible inÂ p , n(1Â p − |u| Hence ω ∈ P φ (G) by Proposition 5.12.

Remark 5.13
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 5.7 (3)), while the idea of the second proof is derived from a general result Proposition 5.12, concerning the decay property (5.7) of this sequence of roots. The inequality (5.10) also allows us to simplify the proof of the main theorem in [17] for the finite group case.