Non-commutative Poisson Boundaries and

Compact Quantum Group Actions

Masaki Izumi

Department of Mathematics, Graduate School of Science

Kyoto University Sakyo-ku, Kyoto 606-8502, Japan

e-mail:izumi@kusm.kyoto-u.ac.jp

and

Mathematical Sciences Research Institute Berkeley, CA 94720, USA

e-mail:izumi@msri.org

December 19, 2000

Dedicated to Yoshiomi Nakagami on the occasion of his sixtieth birthday

Abstract

We discuss some relationship between two different fields, a noncommutative version of the Poisson boundary theory of random walks and the infinite tensor product (abbreviated as ITP) actions of compact quantum groups on von Neumann algebras. In contrast to the ordinary compact group case, the ITP action of a compact quantum group on a factor may allow non-trivial relative commutant of the fixed point subalgebra. We give a probabilistic description of the relative commutant in terms of a non-commutative Markov operator. In particular, we show that the following three objects can be naturally identified in the case of the quantum group SU_{q}(2): (1) the relative commutant of the fixed point algebra under the action, (2) the space of harmonic elements for some non-commutative Markov operator on the dual quantum group of SU_{q}(2), and (3) the weak closure L^{∞}(T\SU_{q}(2)) of one of the Podles quantum spheres. In view of the ordinary Poisson boundary theory of random walks on discrete groups, this shows that symbolically the quantum homogeneous space T\SU_{q}(2) may be regarded as the “Poisson boundary” of a noncommutative random walk on the dual object of SU_{q}(2). An analogy of the Poisson integral formula is also given.

1 Introduction

Group actions on von Neumann algebras and C^{∗}-algebras are of fundamental interest in the theory of operator algebras, and there are a number of fruitful results in history. In particular, several authors have studied compact group actions inspired by the gauge group actions in physics (see for example [1]). Among others, the best understood class so far is that of ITP (infinite tensor product) actions. They are prototypes of so-called minimal actions; an action of a compact group on a factor is said to be minimal if the relative commutant of the fixed point algebra is trivial. For compact group actions, minimality seems to be the right counterpart of freeness [36], which plays a fundamental role in classification of discrete group actions.

The notion of the ITP actions of compact quantum groups was first introduced by Y. Konishi, M. Nagisa, and Y. Watatani in the case of the quantum group SU_{q}(2) for −1 ≤ q ≤ 1, q ≠ 0 acting on the UHF algebra of type 2^{∞} [24]. Konishi showed that the above action on the UHF algebra can be extended to the Powers factor [23], and Y. Nakagami generalized this result to SU_{q}(N) case for general N [29]. For related results, see also [12] [42]. In contrast to the classical case, the above action on the Powers factor is not minimal for q ≠ ±1. This can be observed from a well-known fact in subfactor theory. Namely, if it were minimal, the Jones subfactor of index larger than 4 would be irreducible because it could be obtained via A. Wassermann’s type construction with the above action [43]. However, it is known that the Jones subfactor is reducible if its index is larger than 4 [20]

[31] (see also [18]). The same reasoning using [37] implies that the above mentioned ITP action of SU_{q}(N) is not minimal as well. Note that if one goes beyond the AFD case, there exist several minimal actions of SU_{q}(N) constructed by Y. Ueda [40] [41]. Based on Ueda’s actions, D. Shlyakhtenko and Ueda recently constructed irreducible subfactors of the free group factor L(F_{∞}) for all index values larger than 4 [38]. Whether minimal actions can be realized in the AFD factors or not is still a fascinating open problem.

One of the main purposes of the present work is to investigate mathematical structure behind the above sharp contrast between the ordinary and quantum group cases about minimality. To the best knowledge of the author, this type of difference has never been observed as far as algebraic structure is concerned. Indeed, it turns out that the difference manifests itself only in non-commutative probability theoretical analysis. Symbolically, the relative commutant of the fixed point algebra under the ITP action of a compact quantum group G should be considered as the “L^{∞}-space” over the “Poisson boundary ” of a “random walk” on the dual object of G. More precisely, we can show, under a mild assumption, that the relative commutant is completely positively isometric to the set of harmonic elements H^{∞}(M̂, P_{μ}) of some non-commutative Markov operator P_{μ} on the dual Hopf von Neumann algebra M̂. Another purpose of the present work is to show that there exist rich examples of the Poisson boundary theory of non-commutative Markov operators on discrete quantum groups as well as ordinary random walks on discrete groups [22] [44]. Note that random walks on the dual of SU(N) were discussed by Ph. Biane in [3] [4] [5], and it would be interesting to compare his results with our approach.

In the case of the ITP action of G = SU_{q}(2), further detailed analysis is possible. Namely, we explicitly determine the structure of H^{∞}(M̂, P_{μ}) using the full power of the representation theory of SU_{q}(2). In consequence, we show that the relative commutant of the fixed point algebra is isomorphic to the weak closure L^{∞}(T\SU_{q}(2)) of one of the Podles quantum spheres introduced in [32], both as a von Neumann algebra and as a SU_{q}(2)-comodule. One may be tempted to rephrase this fact as “ the Poisson boundary of the random walk on the dual of SU_{q}(2) can be identified with the quantum homogeneous space T\SU_{q}(2).” We also construct an explicit map from L^{∞}(T\SU_{q}(2)) to H^{∞}(M̂, P_{μ}), which should be interpreted as a { \atop SU_{q}(2)}- version of the Poisson integral formula. The construction of this map makes sense for general compact quantum groups, and it implies that the ITP actions are never minimal for non-Kac compact quantum groups.

Typing of this paper was finished while the author stayed in the Mathematical Sciences Research Institute, and he would like to thank MSRI for their hospitality. The author is grateful to Yoshimichi Ueda for his critical reading of an early version of the draft, and to Shigeru Yamagami and Yoshiomi Nakagami for useful information on compact quantum groups. This work is partly supported by Japanese Ministry of Education, Science, Sports and Culture. The main results of this paper were announced in [17].

2 Preliminaries

In this section, we collect basic notions for compact quantum groups and the Poisson boundaries of non-commutative Markov operators in order to fix notation. Since necessary materials for the subject of the present work are gathered from several different sources, we try to make exposition selfcontained as much as possible. For general theory of compact quantum groups, our standard references are [2] [46]. For more details of the classical theory of the Poisson boundaries of random walks on discrete groups, readers are encouraged to consult [21] [22] [44] and references in them.

Throughout the paper, the symbol ⊗ stands for the minimal tensor product whenever we work on C^{∗}-algebras. For the von Neumann algebra tensor product and weak closure of spatial tensor product of operator systems, we use the symbol ¯⊗. The algebraic tensor product is denoted by ⊗_{alg} when it is necessary to be emphasized. For two densely defined closed operators T_{1} and T_{2} on a Hilbert space, we denote by T_{1} ¯⊗T_{2} the closure of their tensor product T_{1} ⊗_{alg} T_{2} defined on the algebraic tensor product of their domains.

2-1. Compact Quantum Groups. We start with S. L. Woronowicz’ C^{∗}-algebra approach to compact quantum groups.

Let G = (A, U, δ) be a compact quantum group, a matrix pseudogroup in the sense of S. L. Woronowicz [46]. Namely, A is a unital C^{∗}-algebra, and U = (U_{i,j})_{i,j∈I_{U}} is an A-valued unitary matrix in B(ℓ^{2}(I_{U})) ⊗ A with I_{U} a finite index set, such that the set {U_{i,j}}_{i,j∈I_{U}} generate A as a C^{∗}-algebra. The coproduct δ is a ∗-homomorphism δ : A ⟶ A ⊗ A satisfying

δ(U_{i,j}) = ∑_{k∈I_{U}} U_{i,k} ⊗ U_{k,j}.

We set δ^{(1)} = δ and inductively define δ^{(n)} = (id ⊗ δ) · δ^{(n−1)} for every natural number n. Although one can define a compact quantum group without using the “fundamental corepresentation U”, (or in other words, the finite generation condition is not necessary for the general context), we fix U as above in order to introduce the ITP action of the adjoint action of U. The dense ∗-algebra generated by {U_{i,j}}_{i,j∈I_{U}} is denoted by A.

An A-valued unitary matrix w = (w_{i,j})_{i,j∈I_{w}} ∈ B(ℓ^{2}(I_{w})) ⊗ A is called a unitary corepresentation of (A, δ) (called a unitary representation of G in [46]) if the following holds:

δ(w_{i,j}) = ∑_{k∈I_{w}} w_{i,k} ⊗ w_{k,j}.

Usual notions such as equivalence, direct sums, and irreducibility of unitary corepresentations were established in [46]. We denote by Ĝ the set of equivalence classes of finite dimensional irreducible unitary corepresentations. For each s ∈ Ĝ, we fix a representative w(s) = (w(s)_{i,j})_{i,j∈I_{s}} ∈ B(ℓ^{2}(I_{s})) ⊗ A. Let W (s) be the linear span of {w(s)_{i,j}}_{i,j∈I_{s}}, which depends only on s ∈ Ĝ. Then as a linear space, A is the direct sum,

A = ⨁_{s∈Ĝ} W (s).  (2.1.1)

The antipode κ, which is an invertible antimultiplicative linear map from A to itself, is defined by

κ(w(s)_{i,j}) = w(s)_{j,i}^{∗}.

The counit ϵ is the character of A determined by

ϵ(w(s)_{i,j}) = δ_{i,j}, s ∈ Ĝ.

Let h be the Haar measure, which is a unique state of A satisfying the following invariance condition:

(h ⊗ id) · δ(x) = (id ⊗ h) · δ(x) = h(x)1, x ∈ A.

On A, h is determined as follows: h(w(s)_{i,j}) = 0 unless s is the trivial representation 0 ∈ Ĝ. We denote by (π_{h}, H_{h}, Ω_{h}) the GNS triple of h, and by Λ_{h} the natural map from A to H_{h}. We assume that π_{h} is faithful, and we will omit π_{h} in what follows. The multiplicative unitary V is defined by the bounded extension of the following operator [2]:

V (Λ_{h}(x) ⊗ ξ) = δ(x)(Ω_{h} ⊗ ξ), x ∈ A, ξ ∈ H_{h}.

V satisfies the following pentagon equation:

V_{12}V_{13}V_{23} = V_{23}V_{12}.

For s ∈ Ĝ, we define a system of matrix units {e(s)_{ij}}_{i,j∈I_{s}} ⊂ B(H_{h}) by

e(s)_{ij}Λ_{h}(w(t)_{k,l}) = δ_{s,t}δ_{j,l}Λ_{h}(w(t)_{k,i}).  (2.1.2)

(Note that the rank of e(s)_{ii} is not 1.) Then, the dual Hopf algebra Â is the C^{∗}-direct sum of the finite dimensional simple C^{∗}-subalgebras {Â(s)}_{s∈Ĝ}, where

Â(s) = span{e(s)_{ij}}_{i,j∈I_{s}}.

Note that we have

V = ∑_{s∈Ĝ} ∑_{i,j∈I_{s}} e(s)_{i,j} ⊗ w(s)_{i,j} ∈ M(Â ⊗ A),  (2.1.3)

where M(Â ⊗ A) stands for the multiplier algebra of Â ⊗ A.

We denote by M and M̂ the weak closure of A and Â respectively. In order to save notations, we use the same symbols δ and h for their weakly continuous extensions to M. Then, δ on M is given by

δ(x) = V (x ⊗ 1)V^{∗}, x ∈ M.  (2.1.4)

The dual coproduct δ̂ on Â (and on M̂) is defined by

δ̂(x) = V^{∗}(1 ⊗ x)V.  (2.1.5)

2-2. Unbounded Realization of the Dual Hopf Algebra. Although it is enough to work on bounded operators in order to formulate compact quantum groups, the dual Hopf algebras realized as unbounded operator algebras are more useful for practical computation (it is easier to construct representations of Lie algebras than Lie groups, of course!). In this subsection, we discuss unbounded realization of the dual Hopf algebra of A.

We denote by A^{∗} the set of (not necessarily bounded) linear functionals on A. Then, A^{∗} has a natural Hopf ∗-algebra structure, and we use the symbol Â when A^{∗} is regarded as a Hopf algebra. The product of two elements ϕ, ψ ∈ Â is given by the following convolution product

(ϕ ∗ ψ)(x) = (ϕ ⊗ ψ) · δ(x), x ∈ A.  (2.2.1)

The antipode κ̂ and the ∗ operation # on ϕ ∈ Â are defined by

κ̂(ϕ)(x) = ϕ(κ(x)), x ∈ A,  (2.2.2)
ϕ^{#}(x) = \overline{ϕ(κ(x)^{∗})}, x ∈ A.  (2.2.3)

We denote the coproduct of Â by δ̂_{0}, which is strictly speaking a map from Â to (A ⊗_{alg} A)^{∗}, and is defined by

δ̂_{0}(ϕ)(x ⊗ y) = ϕ(xy), x, y ∈ A.  (2.2.4)

Note that thanks to (2.1.1) and (2.1.2), Â is naturally identified with the direct product algebra

∏_{s∈Ĝ} Â(s),

which will be further identified with the algebra ˜ M̂ of densely defined closed operators on H_{h} affiliated with M̂ as below.

Let H be the dense subspace Λ_{h}(A) ⊂ H_{h}. For x ∈ A and ϕ ∈ Â, we set

ϕ ∗ x = (id ⊗ ϕ) · δ(x),
x ∗ ϕ = (ϕ ⊗ id) · δ(x).

We introduce a linear operator R_{0}(ϕ) of H_{h}, whose domain is H, by

R_{0}(ϕ)Λ_{h}(x) = Λ_{h}(ϕ ∗ x), x ∈ A.  (2.2.5)

Then, it is easy to show that R_{0}(ϕ) maps H into itself and the closure of R_{0}(ϕ), denoted by R(ϕ), is affiliated with the finite type I von Neumann algebra M̂. Therefore, there is no complicated problem about domains and every formal calculation involving R(Â) can be justified. R_{0} is a representation of the algebra Â satisfying R_{0}(ϕ)^{∗} = R(ϕ^{#}). In the same way, for ψ ∈ (A ⊗_{alg} A)^{∗} we define an operator R_{00}(ψ) of H_{h} ⊗ H_{h}, whose domain is H ⊗_{alg} H, and we denote its closure by R(ψ). Then, it is easy to show that the following holds:

(1 ⊗ R_{0}(ϕ))V ξ = V R_{00}(δ̂_{0}(ϕ))ξ, ξ ∈ H ⊗_{alg} H.

As before, we abuse notation and use the same symbol δ̂ for its natural

extension to ˜ M̂ defined by (2.1.5). Then, the above implies

δ̂(R(ϕ)) = R(δ̂_{0}(ϕ)).

In [46], Woronowicz introduced a unique family {f_{z}}_{z∈C} ⊂ Â of characters, called the Woronowicz characters, satisfying the following properties:

(F1) For every a ∈ A, the mapping C ∋ z ↦ f_{z}(a) is an entire function of exponential growth on the right half-plane.

(F2) f_{z}(1) = 1 for all z ∈ C.

(F3) f_{z} ∗ f_{z′} = f_{z+z′} for all z, z′ ∈ C.

(F4) f_{z}(κ(x)) = f_{−z}(x), f_{z}(x^{∗}) = \overline{f_{−\overline{z}}(x)} for all x ∈ A, z ∈ C.

(F5) κ^{2}(x) = f_{−1} ∗ x ∗ f_{1} for all x ∈ A.

(F6) h(xy) = h(y(f_{1} ∗ x ∗ f_{1})), for all x, y ∈ A.

Note that f_{0} is nothing but the counit by construction in [46].

(F4) implies f_{t}^{#} = f_{t} for real t and that R(f_{t}) = R(f_{\frac{t}{2}} )^{2} is a positive self-adjoint operator for t ∈ R. We set ρ = R(f_{1}), which satisfies R(f_{z}) = ρ^{z} for all z ∈ C. Since f_{1} is a character, we have

δ̂(ρ) = ρ ¯⊗ρ,  (2.2.6)

that is, ρ is a group-like element. (F5) implies

κ̂^{2}(R_{0}(ϕ)) = ρR_{0}(ϕ)ρ^{−1}, ϕ ∈ Â.

The operator ρ^{1/2} plays an important role in their analysis in [47]. ρ will be used to construct invariant states of the ITP actions later.

In the rest of this subsection, we introduce fundamental objects for the Woronowicz algebra structure of M. Since f_{z} is a character, the map

g_{z,w}(x) := f_{z} ∗ x ∗ f_{w}, x ∈ A,

defines an h preserving algebra automorphism of A for each z, w ∈ C. (F4) implies g_{z,w}(x)^{∗} = g_{−\overline{z},−\overline{w}}(x^{∗}), which shows that if z and w are pure imaginary, g_{z,w} extends to a ∗-automorphism of A and also M. Among others, there are two distinguished one parameter automorphism groups in this family of automorphisms: namely the modular automorphism group {σ_{t}^{h}}_{t∈R} of h and the deformation automorphism group {τ_{t}}_{t∈R} introduced in [26]. They are given by

σ_{t}^{h}(x) = f_{it} ∗ x ∗ f_{it}, x ∈ A,  (2.2.7)
τ_{t}(x) = f_{−it} ∗ x ∗ f_{it}, x ∈ A.  (2.2.8)

In a similar way using (F4) and (F5), we can introduce an involutive ∗- antiautomorphism j of A and also M [2, Section 5], called unitary antipode, by

j(x) = f_{\frac{1}{2}} ∗ κ(x) ∗ f_{− \frac{1}{2}} , x ∈ A.  (2.2.9)

Note that κ can be recovered from j and τ by

κ(x) = j · τ_{− \frac{i}{2}} (x) = τ_{− \frac{i}{2}} · j(x), x ∈ A.  (2.2.10)

2-3. Modules and Comodules. Let K be a linear space. K is called a right comodule of A if there exists a linear map α : K ⟶ K ⊗ A satisfying

(α ⊗ id) · α = (id ⊗ δ) · α,

where the tensor product should be understood in an appropriate way depending on which category we are working on. For s ∈ Ĝ, the spectral subspace of K for s is defined to be

{ξ ∈ K; α(ξ) ∈ K ⊗ W (s)}.

Left comodules are also define in a similar way, and comodules of A, M, Â, Â, and M̂ are defined in the same way.

Let w = (w_{i,j})_{i,j∈I_{w}} ∈ B(ℓ^{2}(I_{w})) ⊗ A be a unitary corepresentation, and {ξ_{i}}_{i∈I_{w}} be the canonical orthonormal basis of ℓ^{2}(I_{w}). Then, ℓ^{2}(I_{w}) is a right A comodule with α given by

α(ξ_{i}) = ∑_{j∈I_{w}} ξ_{j} ⊗ w_{j,i}.

ℓ^{2}(I_{w}) naturally has left Â module structure given by

π_{w}(ϕ)ξ = (id ⊗ ϕ) · α(ξ), ξ ∈ ℓ^{2}(I_{w}).  (2.3.1)

We also regard π_{w} as a representation of M̂ in an obvious way. When w = w(s) for some s ∈ Ĝ, we denote π_{w} by π_{s} for simplicity, which is identified with the irreducible representation of M̂ corresponding to the simple component Â(s). For two unitary corepresentations w_{1} and w_{2}, the tensor product representation (π_{w_{1}} ⊗ π_{w_{2}}) · δ̂ is simply denoted by π_{w_{1}} ⊗ π_{w_{2}}, and the n-fold tensor product representation of π_{w} is denoted by π_{w}^{⊗n}.

For s, t, r ∈ Ĝ, let N_{s,t}^{r} be the multiplicity of π_{r} in π_{s} ⊗ π_{t}. The quantum dimension d(s) of s is defined by

d(s) = Tr(π_{s}(ρ)),

where Tr stands for the usual trace. Then, Ĝ has fusion algebra structure in the sense of [16] with the structure constant N_{s,t}^{r} and the dimension function d(s).

For a unitary corepresentation w, we introduce a positive linear functional T_{w} ∈ M̂_{∗}^{+} by

T_{w}(x) = Tr(π_{w}(ρx)), x ∈ M̂.

T_{w} is sometimes called a q-trace. (2.2.6) implies

T_{w_{1}} ∗ T_{w_{2}}(x) := (T_{w_{1}} ⊗ T_{w_{2}}) · δ̂(x) = Tr((π_{w_{1}} ⊗ π_{w_{2}})(ρx)),

and in particular,

T_{s} ∗ T_{t} = ∑_{r∈Ĝ} N_{s,t}^{r}T_{r}, s, t ∈ Ĝ.  (2.3.2)

2-4 Actions. Let B be a C^{∗}-algebra. A right (respectively left) action α of (A, δ) on B is a ∗-homomorphism from B to B ⊗ A (respectively A ⊗ B) satisfying the right (respectively left) comodule property above. The fixed point subalgebra of B under α is defined to be

B^{α} = {x ∈ B; α(x) = x ⊗ 1}.

A linear functional φ ∈ B^{∗} is said to be invariant under α if the following condition is satisfied:

(φ ⊗ id) · α(x) = φ(x)1, x ∈ B.

We also define actions of (Â, δ̂), (M, δ), and (M̂, δ̂) in the same way except that in the M and M̂ cases we assume that B is a von Neumann algebra and α is normal.

The simplest example of an action of (A, δ) is δ acting on A from either left or right. Using the same equations (2.1.4) and (2.1.5) and the pentagon equation, we can introduce a right action of (M, δ) on M̂ and a left action of (M̂, δ̂) on M, which will be denoted by Φ and Φ̂ respectively.

Now, we introduce the adjoint actions of unitary corepresentations.

Lemma 2.1 Let w be a unitary corepresentation of (A, δ) and z be an element of the center Z(M̂) of M̂. Then, (1) For x ∈ B(ℓ^{2}(I_{w})) we set

α_{w}(x) = w(x ⊗ 1)w^{∗}.

Then, α_{w} is a right action of (A, δ) on B(ℓ^{2}(I_{w})). The linear span of α_{w}(B(ℓ^{2}(I_{w})))(C ⊗ A) coincides with B(ℓ^{2}(I_{w})) ⊗ A.

(2) The fixed point algebra B(ℓ^{2}(I_{w}))^{α_{w}} is given by

B(ℓ^{2}(I_{w}))^{α_{w}} = π_{w}(M̂)′.

(3) For x ∈ M̂, we have

α_{w} · π_{w}(x) = (π_{w} ⊗ id) · Φ(x).

(4) We define a linear functional φ ∈ B(ℓ^{2}(I_{w}))^{∗} by

φ_{w,z}(x) = Tr(π_{w}(zρ)x), x ∈ B(ℓ^{2}(I_{w})).

Then, φ_{w,z} is invariant under α_{w}.

Proof. (1) It is easy to show that α_{w} is an action. Let {e_{ij}}_{i,j∈I_{w}} be the canonical system of matrix units of B(ℓ^{2}(I_{w})). Then,

α_{w}(e_{ij}) = ∑_{k,l∈I_{w}} e_{kl} ⊗ w_{k,i}w_{l,j}^{∗}.

Since the matrix (w_{p,q}^{∗})_{p,q} is invertible [46], we have

B(ℓ^{2}(I_{w})) ⊗ C ⊂ α_{w}(B(ℓ^{2}(I_{w})))(C ⊗ A),

which proves the statement. (2) and (3) are easy consequences of the definitions. (4) Since π_{w} is a direct sum of irreducible representations, a simple argument shows that it suffices to prove the statements for irreducible w. Thus, we assume z = 1. Then

(φ_{w,1} ⊗ id) · α_{w}(x) = ∑_{i,j,k,l∈I_{w}} Tr(xe_{lk}π_{w}(ρ)e_{ij})w_{i,j}w_{k,l}^{∗}
= ∑_{i,j,k,l∈I_{w}} Tr(xe_{lj})f_{1}(w_{k,i})w_{i,j}w_{k,l}^{∗}.

Thus, (F5) implies

(φ_{w,1} ⊗ id) · α_{w}(x) = ∑_{m,j,k,l∈I_{w}} Tr(xe_{lj})f_{1}(w_{m,j})κ^{2}(w_{k,m})w_{k,l}^{∗}
= ∑_{m,j,k,l∈I_{w}} Tr(xe_{lj})f_{1}(w_{m,j})κ(w_{l,k}w_{m,k}^{∗})
= ∑_{j,l∈I_{w}} Tr(xe_{lj})f_{1}(w_{l,j})1
= Tr(xπ_{w}(ρ))1.

Q.E.D.

2-5. The Poisson Boundaries. In view of the Poisson boundary theory of random walks [21] [22], we adopt the following algebraic definition of the Poisson boundaries, which makes sense also in the non-commutative case.

Let N be a von Neumann algebra. A unital complete positive map P from N to itself is called a non-commutative Markov operator. We say that x ∈ N is P -harmonic if x is fixed by P. We denote by H^{∞}(N, P) the set of P -harmonic elements. Although H^{∞}(N, P) is not an algebra in general, it is a weakly closed operator system [8]: namely it is a unital self-adjoint subspace of N. We show that one can introduce a new product in H^{∞}(N, P) so that it becomes a von Neumann algebra.

We fix a free ultrafilter ω ∈ βN \ N and define a norm one projection E_{ω} from N to H^{∞}(N, P) by the weak limit

E_{ω}(x) = w − lim_{n→ω} \frac{1}{n} ∑_{k=0}^{n−1} P^{k}(x).

Then, the Choi-Effros product E_{ω}(xy) for x, y ∈ H^{∞}(N, P) gives H^{∞}(N, P) abstract von Neumann algebra structure [8]. We call the resulting von Neumann algebra the non-commutative Poisson boundary of (N, P) for the reason stated below. Note that the Choi-Effros product does not depend on ω because every completely positive isometry between two von Neumann algebras is actually an isomorphism. For a normal state φ of N, the restriction of φ to H^{∞}(N, P) induces a normal state of H^{∞}(N, P) considered as a von Neumann algebra, which is called the harmonic state for the initial distribution φ.

To justify the terminology, at this point we give a classical example, which in fact inspired the present work. Let Γ be a discrete group and μ be a probability measure on Γ. Then, the right convolution operator P_{μ} by μ is a Markov operator of the von Neumann algebra N = ℓ^{∞}(Γ). In this case, H^{∞}(N, P) is nothing but the bounded harmonic function space H^{∞}(Γ, μ) and the resulting von Neumann algebra is isomorphic to the L^{∞}- space over the Poisson boundary, (in fact this is even one of the equivalent definitions of the Poisson boundary adopted in [22]). Let ν be an initial distribution and (Ω, ϵ) be the Poisson boundary, where Ω is a Borel space and ϵ is the harmonic measure for the initial distribution ν. Though the Poisson boundary is usually constructed in a purely measure theoretic way, the functional analytic characterization is the following: under some mild assumption on μ and ν, there exists a surjective positive isometry

θ : H^{∞}(Γ, μ) ⟶ L^{∞}(Ω, ϵ),

which intertwines the left actions of Γ. The product of the right-hand side is recovered from the strong limit

θ^{−1}(θ(f_{1})θ(f_{2})) = s − lim_{n→∞} P_{μ}^{n}(f_{1}f_{2}), f_{1}, f_{2} ∈ H^{∞}(Γ, μ).

For f ∈ H^{∞}(Γ, μ), the image θ(f) is interpreted as the restriction of f to the Poisson boundary. The harmonic measure ϵ is, by definition, determined by the following relation

∫_{Γ} f(g)dν(g) = ∫_{Ω} θ(f)(ω)dϵ(ω).

The classical Poisson integral formula on the unit disc is a means to recover the values of a harmonic function f from its boundary values. In the discrete group situation, a concrete realization of the inverse of θ should be considered as its counterpart. When the initial distribution is concentrated on the neutral element e, it takes a very simple form, which is even carried to the case of the dual of SU_{q}(2) (see Section 5). For f ∈ H^{∞}(Γ, μ) and g ∈ Γ, we set f_{g}(h) = f(gh), which is a harmonic function again. Then,

f(g) = f_{g}(e) = ∫_{Ω} θ(f_{g})(ω)dϵ(ω) = ∫_{Ω} θ(f)(gω)dϵ(ω)
= ∫_{Ω} θ(f)(ω)\frac{dϵ(g^{−1}·)}{dϵ}(ω)dϵ(ω).

This is regarded as the discrete group version of the Poisson integral formula and the Radon-Nikodym derivative should be considered as the Poisson kernel.

Now we introduce a family of non-commutative Markov operators on M̂, which generalize the convolution operators on discrete groups.

Lemma 2.2 For s ∈ Ĝ, we define a normal completely positive map Q_{s} from M̂ to itself by

Q_{s}(x) = (id ¯⊗T_{s}) · δ̂(x), x ∈ M̂.

Then,

(1) The family {Q_{s}}_{s∈Ĝ} satisfies

Q_{s} · Q_{t} = ∑_{r∈Ĝ} N_{s,t}^{r}Q_{r}, s, t ∈ Ĝ.

(2) We set

P_{s} = \frac{1}{d(s)}Q_{s}.

For a probability measure μ on Ĝ, we define a non-commutative Markov operator P_{μ} by

P_{μ} = ∑_{s∈Ĝ} μ(s)P_{s}.

Then, we have

P_{μ} · P_{ν} = P_{μ∗ν},

where μ∗ν is the convolution product of two probability measures μ and ν on the fusion algebra Ĝ introduced in [16].

(3) For every x ∈ M̂, we have

(P_{μ} ¯⊗id)(Φ(x)) = Φ(P_{μ}(x)).

(4) For every x ∈ M̂, we have

(id ¯⊗P_{μ})(δ̂(x)) = δ̂(P_{μ}(x)).

Proof. (1) and (2) are direct consequences of (2.3.2), and (4) is immediate from the definitions. It suffices to show (3) for Q_{s} instead of P_{μ}. Let x ∈ M̂. Then,

(Q_{s} ¯⊗id)(Φ(x)) = (id ¯⊗T_{s} ¯⊗id) · (δ̂¯⊗id) · Φ(x)
= (id ¯⊗T_{s} ¯⊗id) · (Ad(V_{12}^{∗}V_{23})(1 ⊗ x ⊗ 1))
= (id ¯⊗T_{s} ¯⊗id) · (Ad(V_{13}V_{23}V_{12}^{∗})(1 ⊗ x ⊗ 1))
= V (id ¯⊗T_{s} ¯⊗id) · (id ¯⊗Φ) · (δ̂¯⊗id)(x)V^{∗}.

Thanks to Lemma 2.1, we have

(T_{s} ¯⊗id) · Φ(y) = T_{s}(y)1, y ∈ M̂.

Thus,

(Q_{s} ¯⊗id)(Φ(x)) = V ((id ¯⊗T_{s})(δ̂(x)) ⊗ 1)V^{∗} = Φ(Q_{s}(x)),

which proves the statement. Q.E.D.

3 ITP Actions

In this section, we show that as in the classical case, the von Neumann algebra structure of the space of harmonic elements H^{∞}(M̂, P_{μ}) has a natural concrete realization for every probability measure μ on Ĝ whose support is a finite generating set. The relative commutant of the fixed point subalgebra of the ITP action on the Powers factor (or the Araki-Woods factor) turns out to be the right realization.

First, we introduce the ITP actions in the C^{∗}-level following [24]. We concentrate on the ITP action of the adjoint action of the “fundamental corepresentation” U = (U_{i,j})_{i,j∈I_{U}} ∈ B(ℓ^{2}(I_{U})) ⊗ A (otherwise, we would consider the C^{∗}-subalgebra generated by the corepresentation we would start with.) Let B_{n} be the tensor product of n copies of B(ℓ^{2}(I_{U})), and B be the UHF algebra generated by the union ∪_{n}B_{n}, namely

B_{n} = ⨂_{k=1}^{n} B(ℓ^{2}(I_{U})),

B = ⨂_{k=1}^{∞} B(ℓ^{2}(I_{U})).

Let {e_{ij}^{k}}_{i,j∈I_{U}} be the canonical system of matrix units of the k-th tensor component, and we set

v_{k} = ∑_{i,j∈I_{U}} e_{ij}^{k} ⊗ U_{i,j} ∈ B_{k} ⊗ A ⊂ B ⊗ A.

Let V_{n} = v_{1}v_{2} ⋯ v_{n}, which is nothing but (π_{U}^{⊗n} ⊗ id)(V). A simple argument using (1) of Lemma 2.1 implies the following:

Lemma 3.1 Under the above notation, we have

(1) For every x ∈ B, the following limit exists in norm topology:

lim_{n→∞} V_{n}(x ⊗ 1)V_{n}^{∗}.

We denote by α(x) the above limit. Then, α is a right (A, δ) action of B.

(2) The linear span of α(B)(C ⊗ A) is dense in B ⊗ A.

We slightly generalize the construction of the invariant states of the ITP actions in [23] [29] [42], which is probably well-known to (even non) specialists. We fix a positive central element z ∈ Z(M̂) such that π_{U}(z) is invertible and T_{U}(z) = 1. As in Subsection 2-4, we define a faithful state φ_{0} of B(ℓ^{2}(I_{U})) by

φ_{0}(x) = Tr(π_{U}(zρ)x), x ∈ B(ℓ^{2}(I_{U})).

We set φ to be the infinite tensor product state of φ_{0},

φ = ⨂_{k=1}^{∞} φ_{0}.

Let (π_{φ}, H_{φ}, Ω_{φ}) be the GNS triple of φ, and Λ_{φ} be the natural map from A to H_{φ}. Since π_{φ} is faithful, we omit the symbol π_{φ} in what follows. Let N be the weak closure of A. We use the same symbol φ for their normal extensions to N.

The following lemma is standard. (See [11, Théorèm 2.9] for more general statements.)

Lemma 3.2 With the above notation, the following hold:

(1) φ is invariant under α.

(2) We define an isometry V_{∞} ∈ B(H_{φ} ⊗ H_{h}) by the continuous extension of the following operator:

V_{∞}(Λ_{φ}(x) ⊗ ξ) = α(x)(Ω_{φ} ⊗ ξ), x ∈ A, ξ ∈ H_{h}.

Then, V_{∞} is a unitary.

(3) α extends to an action of (M, δ) on N.

(4) Let {σ_{t}^{φ}}_{t∈R} be the modular automorphism group of φ, and {τ_{t}}_{t∈R} be the deformation automorphism group of M. Then,

α · σ_{t}^{φ} = (σ_{t}^{φ} ¯⊗τ_{−t}) · α, t ∈ R.

(5) Let J_{φ} be the modular conjugation of N with respect to Ω_{φ}, and Ĵ be the antiunitary operator defined by

ĴxΩ_{h} = j(x^{∗})Ω_{h}, x ∈ M,

which is in fact the modular conjugation of M̂ [26]. Then,

(J_{φ} ⊗ Ĵ)V_{∞}(J_{φ} ⊗ Ĵ) = V_{∞}^{∗}.

Proof. (1) follows from Lemma 2.1 and (2) follows from the previous lemma. Using V_{∞}, α can be expressed as

α(x) = V_{∞}(x ⊗ 1)V_{∞}^{∗}, x ∈ B,

which shows (3).

(4) Let x ∈ B_{n}. Since (⊗_{k=1}^{n}π_{U}(z)) ⊗ 1 commutes (π_{U}^{⊗n} ⊗ id)(V), we have

(σ_{−t}^{φ} ⊗ id) · α · σ_{t}^{φ}(x) = Ad(π_{U}^{⊗n} ⊗ id)((ρ^{−it} ⊗ 1)V (ρ^{it} ⊗ 1))(x ⊗ 1).

We claim that the following holds:

((ρ^{it} ⊗ 1)V (ρ^{−it} ⊗ 1)) = (id ⊗ τ_{t})(V).

Indeed, for s ∈ Ĝ we have

∑_{k,l∈I_{s}} ρ^{it}e(s)_{kl}ρ^{−it} ⊗ w(s)_{k,l} = ∑_{k,l,p,q,} e(s)_{pq} ⊗ f_{it}(w(s)_{p,k})f_{−it}(w(s)_{l,q})w(s)_{k,l}
= ∑_{p,q,} e(s)_{pq} ⊗ τ_{t}(w(s)_{p,q}).

Thus, (2.1.3) and (2.2.7) imply the claim, and so,

(σ_{−t}^{φ} ⊗ id) · α · σ_{t}^{φ}(x) = (id ⊗ τ_{−t}) · α(x).

Since ∪_{n=1}^{∞}B_{n} is dense in M, we get the statement.

(5) Let x, y ∈ B_{n} and a, b ∈ A. Note that x and y are analytic elements for σ^{φ}, and so are a and b for σ^{h} and τ^{h}. On one hand, we have

⟨(J_{φ} ⊗ Ĵ)V_{∞}(J_{φ} ⊗ Ĵ)(xΩ_{φ} ⊗ aΩ_{h}), yΩ_{φ} ⊗ bΩ_{h}⟩
= ⟨σ_{− \frac{i}{2}}^{φ} (y^{∗})Ω_{φ} ⊗ j(b^{∗})Ω_{h}, V_{∞}(σ_{− \frac{i}{2}}^{φ} (x^{∗})Ω_{φ} ⊗ j(a^{∗})Ω_{h})⟩
= ⟨σ_{− \frac{i}{2}}^{φ} (y^{∗})Ω_{φ} ⊗ j(b^{∗})Ω_{h}, α(σ_{− \frac{i}{2}}^{φ} (x^{∗}))(Ω_{φ} ⊗ j(a^{∗})Ω_{h})⟩
= ⟨σ_{− \frac{i}{2}}^{φ} (y^{∗})Ω_{φ} ⊗ j(b^{∗})Ω_{h}, (σ_{− \frac{i}{2}}^{φ} ⊗ τ_{\frac{i}{2}} )(α(x^{∗}))(Ω_{φ} ⊗ j(a^{∗})Ω_{h})⟩
= (φ ⊗ h)((1 ⊗ j(a))(σφ_{\frac{i}{2}} ⊗ τ_{− \frac{i}{2}} )(α(x))(σ_{− \frac{i}{2}}^{φ} (y^{∗}) ⊗ j(b^{∗})))
= (φ ⊗ h)((σφ_{\frac{i}{2}}(y^{∗}) ⊗ j(aσ_{−i}^{h}(b^{∗})))(σφ_{\frac{i}{2}} ⊗ τ_{− \frac{i}{2}} )(α(x)))
= (φ ⊗ h)((y^{∗} ⊗ τ_{\frac{i}{2}} · j(aσ_{−i}^{h}(b^{∗})))α(x))
= (φ ⊗ h)((y^{∗} ⊗ κ^{−1}(aσ_{−i}^{h}(b^{∗}))α(x))
= (φ ⊗ h)((y^{∗} ⊗ 1)(id ⊗ κ)(α(x))(1 ⊗ aσ_{−i}^{h}(b^{∗}))).

In the above, we used the result of (4), the KMS condition of σ^{φ} and σ^{h}, j · σ_{t}^{h} = σ_{−t}^{h} · j, (2.2.10), and the fact that κ preserves h. On the other hand

⟨V^{∗}(xΩ_{φ} ⊗ aΩ_{h}), yΩ_{φ} ⊗ bΩ_{h}⟩ = (φ ⊗ h)((1 ⊗ b^{∗})α(y^{∗})(x ⊗ a))
= (φ ⊗ h)(α(y^{∗})(x ⊗ aσ_{−i}^{h}(b^{∗}))).

Therefore, it suffices to show the following for x, y ∈ B_{n}:

(φ ⊗ κ)((y ⊗ 1)α(x)) = (φ ⊗ id)(α(y)(x ⊗ 1)).

Using the KMS condition of σ^{φ} and the property (F5) of the Woronowicz characters, we get

(φ ⊗ id)(α(y)(x ⊗ 1))

= ∑_{s∈Ĝ} ∑_{k,l,p,q∈I_{s}} φ(π_{U}^{⊗n}(e(s)_{kl})yπ_{U}^{⊗n}(e(s)_{qp})x)w(s)_{k,l}w(s)_{p,q}^{∗}
= ∑_{s∈Ĝ} ∑_{k,l,p,q∈I_{s}} φ(yπ_{U}^{⊗n}(e(s)_{qp})xσ_{−i}^{φ}(π_{U}^{⊗n}(e(s)_{kl})))w(s)_{k,l}w(s)_{p,q}^{∗}
= ∑_{s∈Ĝ} ∑_{k,l,p,q∈I_{s}} φ(yπ_{U}^{⊗n}(e(s)_{qp})xπ_{U}^{⊗n}(ρe(s)_{kl}ρ^{−1}))w(s)_{k,l}w(s)_{p,q}^{∗}
= ∑_{s∈Ĝ} ∑_{i,j,p,q∈I_{s}} φ(yπ_{U}^{⊗n}(e(s)_{qp})xπ_{U}^{⊗n}(e(s)_{ij}))(f_{−1} ∗ w(s)_{i,j} ∗ f_{1})w(s)_{p,q}^{∗}
= ∑_{s∈Ĝ} ∑_{i,j,p,q∈I_{s}} φ(yπ_{U}^{⊗n}(e(s)_{qp})xπ_{U}^{⊗n}(e(s)_{ij}))κ^{2}(w(s)_{i,j})κ(w(s)_{q,p})
= ∑_{s∈Ĝ} ∑_{i,j,p,q∈I_{s}} φ(yπ_{U}^{⊗n}(e(s)_{qp})xπ_{U}^{⊗n}(e(s)_{ij}))κ(w(s)_{q,p}w(s)_{j,i}^{∗})
= (φ ⊗ κ)((y ⊗ 1)α(x)),

which finishes the proof. Q.E.D.

We use the same symbol α for their extensions to N. Let R = N ∩ N^{α}′. Then, we have

α(R) ⊂ (N ¯⊗M) ∩ (N^{α}′ ¯⊗C) = R ¯⊗M.

Thus, α induces an action of (M, δ) on R. We denote by α_{R} and φ_{R} the restriction of α and φ to R respectively. The following lemma shows that R has enough spectral subspaces for the (M, δ) action:

Lemma 3.3 Let R, α_{R}, and φ_{R} be as above. Then,

(1) The linear span of α(R)(C ⊗ M) is weakly dense in R ¯⊗M.

(2) For s ∈ Ĝ, we set

R(s) = {x ∈ R; α(x) ∈ R ⊗ W (s)}.

Then, the linear span of ∪_{s∈Ĝ}R(s) is weakly dense in R.

Proof. (1) Thanks to (4) of Lemma 3.2, the fixed point algebra is globally invariant under the modular automorphism group of φ, and so is the relative commutant. Therefore, Takesaki’s theorem [39] assures that there exists a unique φ preserving normal conditional expectation E_{R} from M onto R. We claim that E_{R} commutes with α, that is, α · E_{R} = (E_{R} ¯⊗id) · α. Indeed, let e_{R} be the projection onto the closure of RΩ_{φ}. α(R) ⊂ R ¯⊗M implies

V_{∞}(e_{R} ⊗ 1) = (e_{R} ⊗ 1)V_{∞}(e_{R} ⊗ 1).

Since e_{R} commutes with J_{φ}, (4) of Lemma 3.2 implies that e_{R} ⊗ 1 commutes with V_{∞}. Thus, for x ∈ N,

(E_{R} ¯⊗id) · α(x)(Ω_{φ} ⊗ Ω_{h}) = (e_{R} ⊗ 1)V_{∞}(xΩ_{φ} ⊗ Ω_{h}) = V_{∞}(e_{R}xΩ_{φ} ⊗ Ω_{h})
= α(E_{R}(x))(Ω_{φ} ⊗ Ω_{h}).

Since Ω_{φ} ⊗ Ω_{h} is separating for N ¯⊗M, we get the claim. Thanks to (2) of Lemma 3.1, the linear span of α(N)(C ⊗ M) is weakly dense in N ¯⊗M. Applying E_{R} ¯⊗id to both sides, we get the statement.

(2) Since the Haar measure h is faithful on M and A is a dense ∗- subalgebra, the Hahn-Banach separating theorem implies that the following set is dense in the predual M_{∗}:

{h(·a); a ∈ A}.

On the other hand, as in [33, Theorem 1.5], for s ∈ Ĝ we have

(id ¯⊗h)(α(x)(1 ⊗ w(s)_{i,j}^{∗})) ∈ R(s), x ∈ R.

Therefore, (1) implies the statement. Q.E.D.

Let n_{s} be the multiplicity of w(s), s ∈ Ĝ in U. Since z is a positive central element with T_{U}(z) = 1, there exist positive numbers {z_{s}}_{s∈Ĝ} such that

z = ∑_{s∈Ĝ} z_{s}1_{s},

1 = ∑_{s∈Ĝ} z_{s}n_{s}d(s),

where 1_{s} is the unit of Â(s). We introduce a probability measure μ on Ĝ by

μ = ∑_{s∈Ĝ} z_{s}n_{s}d(s)δ_{s},

where δ_{s} is the Dirac measure at s ∈ Ĝ. We say that μ is generating if {U_{i,j}}_{i,j∈I_{U}} generate A (not only as a ∗-algebra) as an algebra. The goal of this section is to give a precise description of the triple (R, α_{R}, φ_{R}) in terms of the space of P_{μ}-harmonic elements H^{∞}(M̂, P_{μ}). As in the classical case, the key tool of our analysis is the non-commutative martingale convergence theorem (see [9, Theorem 2] for the proof).

Lemma 3.4 Let E_{n} be the unique φ preserving conditional expectation from N to B_{n}. Then,

(1) For every element x ∈ N, the sequence {E_{n}(x)}_{n=1} converges to x in strong ∗ topology.

(2) On the other hand, if {x_{n}}_{n=1}^{∞} ⊂ N is a bounded sequence satisfying E_{n}(x_{n+1}) = x_{n} for every n ∈ N, then there exists a unique element x ∈ N such that E_{n}(x) = x_{n}.

Note that E_{n} is given by an obvious slice map in the above. For a natural number n, we set

Â_{n} = ⨁_{s∈supp(μ^{n})} Â(s),

where μ^{n} stands for the n-fold convolution of μ [16] and supp(μ^{n}) stands for the support of μ^{n}. Let F_{n} be the natural projection from M̂ onto Â_{n}. Then π_{U}^{⊗n} is injective on Â_{n} and the kernel of π_{U}^{⊗n} coincides with that of F_{n}. Let x ∈ R. Since x commutes with B_{n}^{α}, we have E_{n}(x) ∈ B_{n}^{α}′ ∩ B_{n}. (2) of Lemma 2.1 shows that E_{n}(x) belongs to the image of π_{U}^{⊗n}, and so there exists a unique element y_{n} ∈ Â_{n} satisfying π_{U}^{⊗n}(y_{n}) = E_{n}(x). Therefore, all information about x is encoded in the sequence {y_{n}}_{n=1}^{∞}. This is so called a time dependent harmonic sequence [21], and in a good situation, it actually comes from a single harmonic element.

Lemma 3.5 Under the above notation, we have

(1) For every natural number n, the following holds:

π_{U}^{⊗n} · P_{μ} = E_{n} · π_{U}^{⊗(n+1)}.

(2) We assume that for every s ∈ Ĝ, there exists some integer k such that π_{s} is contained in π_{U}^{⊗k}, or in other words, μ is a generating measure. Then, for every bounded sequence {y_{n}}_{n=1}^{∞} ⊂ M̂ satisfying y_{n} ∈ Â_{n} and F_{n} · P_{μ}^{m}(y_{n+m}) = y_{n} for m, n ∈ N, there exists a unique P_{μ}-harmonic element y ∈ H^{∞}(M̂, P_{μ}) such that y_{n} = F_{n}(y).

Proof. (1) Using P_{μ} = (id ¯⊗Tr(π_{U}(zρ·))) · δ, we get

E_{n} · π_{U}^{⊗(n+1)} = (π_{U}^{⊗n} ¯⊗Tr(π_{U}(zρ·))) · δ = π_{U}^{⊗n} · P_{μ}.

(2) We essentially follow the argument in [21]. Let p be the smallest natural number such that supp(μ^{n+p}) ∩ supp(μ^{n}) is not empty for some n. Such p always exists because there exists m ∈ N such that π_{U}^{⊗m} contains the trivial representation 0 ∈ Ĝ and supp(μ) ∩ supp(μ^{m+1}) is not empty. For each i = 0, 1, ⋯ p − 1, we set

Ĝ_{i} = ⋃_{n∈N} supp(μ^{i+np}).

By assumption, we have

Ĝ = ⋃_{i=1}^{m} Ĝ_{i}.

Now we apply Foguel’s theorem [13] to μ^{p}, (which was originally proven for discrete groups but still works for fusion algebras), and get the following ℓ^{1} norm estimate:

lim_{n→∞} ||μ^{p+n} − μ^{n}||_{1} = 0.

This implies

lim_{n→∞} ||P_{μ}^{p+n} − P_{μ}^{n}|| = 0.

For every natural number k, we have

||F_{k}(y_{p+k}) − y_{k}|| = lim_{n→∞} ||F_{k} · P_{μ}^{n}(y_{p+k+n}) − F_{k} · P_{μ}^{p+n}(y_{p+k+n})|| = 0.

Thus, there exist P_{μ}^{m}-harmonic elements y_{i}^{0}, i = 0, 1, ⋯, p − 1 in

( ⋃_{s∈Ĝ_{i}} Â_{s})′′,

such that y_{i+np} = F_{i+np}(y_{i}^{0}) for all n ∈ N. F_{n} · P_{μ}(y_{n+1}) = y_{n} implies

P_{μ}(y_{i+1}^{0}) = y_{i}^{0}, i = 0, 1, ⋯ p − 1,

where y_{p}^{0} is understood as y_{0}^{0}.

We claim that Ĝ_{i} and Ĝ_{j} are disjoint for 0 ≤ i < j ≤ p − 1. Indeed, suppose that Ĝ_{i} ∩ Ĝ_{j} is not empty. Then there exist natural numbers a and b such that supp(μ^{i+ap}) ∩ supp(μ^{j+bp}) is not empty. We take large enough n such that supp(μ^{(n−a)p}) ∩ supp(μ^{(n−b)p}) is not empty. Note that such n exists thanks to the above norm estimate. Then, supp(μ^{i+np}) ∩ supp(μ^{j+np}) is not empty. This contradicts the choice of p and we get the claim. Therefore, y = y_{0}^{0} + y_{1}^{0} + ⋯ + y_{p−1}^{0} is the desired P_{μ}-harmonic element. Q.E.D.

Remark. Wang’s examples of ergodic ITP actions in [42] do not satisfy the assumption of (2). This is the reason our result does not apply to his case.

Now we state our main theorem in this section.

Theorem 3.6 Under the above notation, we assume that for every s ∈ Ĝ, there exists a natural number k such that π_{s} is contained in π_{U}^{⊗k}, or equivalently {U_{i,j}} generate A as an algebra (not just as a ∗-algebra). Then,

(1) For every x ∈ H^{∞}(M̂, P_{μ}), the sequence {π_{U}^{⊗n}(x)}_{n=1}^{∞} converges to an element in R in strong ∗ topology. We denote by θ(x) this limit. Then, θ is a surjective unital completely positive isometry from H^{∞}(M̂, P_{μ}) to R.

(2) For any finite elements x_{1}, x_{2}, ⋯, x_{j} ∈ H^{∞}(M̂, P_{μ}), the sequence {P_{μ}^{n}(x_{1}x_{2} ⋯ x_{j})}_{n=1}^{∞} converges in strong ∗ topology, and

θ( lim_{n→∞} P_{μ}^{n}(x_{1}x_{2} ⋯ x_{j})) = θ(x_{1})θ(x_{2}) ⋯ θ(x_{j}).

This means that θ gives a concrete realization of the non-commutative Poisson boundary of P_{μ}, and in consequence θ is a homeomorphism in σ-weak topology.

(3) Let ϵ̂ be the counit of M̂. Then, we have

φ_{R} · θ(x) = ϵ̂(x), x ∈ H^{∞}(M̂, P_{μ}).

This means that φ_{R} is the harmonic state for the initial distribution ϵ̂.

(4) For s ∈ Ĝ, we set

H^{∞}(M̂, P_{μ})(s) = {x ∈ H^{∞}(M̂, P_{μ}); Φ(x) ∈ H^{∞}(M̂, P_{μ}) ⊗ W (s)}.

Then, the linear span of the union ∪_{s∈Ĝ}H^{∞}(M̂, P_{μ})(s) is weakly dense in H^{∞}(M̂, P_{μ}), and

Φ(H^{∞}(M̂, P_{μ})) ⊂ H^{∞}(M̂, P_{μ}) ¯⊗M.

θ intertwines the (M, δ) actions, namely

(θ ¯⊗id) · Φ|_{H^{∞}(M̂,P_{μ})} = α_{R} · θ.

Proof. (1) Let x ∈ H^{∞}(M̂, P_{μ}). Then {π_{U}^{⊗n}(x)}_{n=1}^{∞} is a bounded sequence in N such that π_{U}^{⊗n}(x) ∈ B_{n} ∩ B_{n}^{α}′ for all n ∈ N. Moreover, (1) of Lemma 3.4 implies

E_{n}(π_{U}^{⊗(n+1)}(x)) = π_{U}^{⊗n}(x).

Therefore, the non-commutative martingale convergence theorem implies that the strong ∗ limit exists. Since θ(x) commutes with B_{n}^{α} for all n, θ(x) ∈ R. Being the limit of unital completely positive maps, θ is unital completely positive. Since {π_{U}^{⊗n}}_{n=1}^{∞} is a faithful family of representations of M̂, we get

||x|| = sup_{n} ||π_{U}^{⊗n}(x)|| = ||θ(x)||.

The same argument for matrices with entries in H^{∞}(M̂, P_{μ}) shows that θ is a complete isometry. Thanks to Lemma 3.5, θ is surjective and θ^{−1} is also completely positive.

(2) Let x_{1}, x_{2}, ⋯, x_{j} ∈ H^{∞}(M̂, P_{μ}). Then, the non-commutative martingale convergence theorem implies

θ(x_{1})θ(x_{2}) ⋯ θ(x_{k}) = lim_{n→∞} E_{n}(θ(x_{1})θ(x_{2}) ⋯ θ(x_{k}))
= lim_{n→∞} lim_{m→∞} E_{n}(π_{U}^{⊗m}(x_{1})π_{U}^{⊗m}(x_{2}) ⋯ π_{U}^{⊗m}(x_{k}))
= lim_{n→∞} lim_{m→∞} E_{n}(π_{U}^{⊗m}(x_{1}x_{2} ⋯ x_{k}))
= lim_{n→∞} lim_{m→∞} π_{U}^{⊗n}(P_{μ}^{m−n}(x_{1}x_{2} ⋯ x_{k}))

where every limit is taken in strong ∗ topology. In particular, for every fixed n the following limit exists

lim_{m→∞} F_{n}(P_{μ}^{m}(x_{1}x_{2} ⋯ x_{k})).

Since {P_{μ}^{m}(x_{1}x_{2} ⋯ x_{k})}_{m=1}^{∞} is a bounded sequence, it converges in strong ∗ topology and we get the statement.

(3) For x ∈ H^{∞}(M̂, P_{μ}) we have

φ_{R}(θ(x)) = φ(E_{1}(θ(x))) = φ(π_{U}(x)) = ϵ̂(P_{μ}(x)) = ϵ̂(x),

which shows the statement.

(4) Let x ∈ H^{∞}(M̂, P_{μ}) and ψ ∈ M_{∗}. Then, thanks to (3) of Lemma 2.2, (id ¯⊗ψ)(Φ(x)) is P_{μ}-harmonic. (3) of Lemma 2.1 implies

θ((id ¯⊗ψ)(Φ(x))) = lim_{n→∞}(π_{U}^{⊗n} ¯⊗ψ)(Φ(x)) = lim_{n→∞}(id ¯⊗ψ)(α(π_{n}^{⊗n}(x)))
= (id ¯⊗ψ)(α_{R}(θ(x))).

Thus, (2) of Lemma 3.5 and the above (2) imply that the preimage of R(s) by θ is H^{∞}(M̂, P_{μ})(s) and the linear span of the union ∪_{s∈Ĝ}H^{∞}(M̂, P_{μ})(s) is dense in H^{∞}(M̂, P_{μ}) in σ-weak topology, which implies

Φ(H^{∞}(M̂, P_{μ})) ⊂ H^{∞}(M̂, P_{μ}) ¯⊗M.

Therefore, (θ ¯⊗id) · Φ|_{H^{∞}(M̂,P_{μ})} makes sense and

(θ ¯⊗id) · Φ|_{H^{∞}(M̂,P_{μ})} = α_{R} · θ.

Q.E.D.

Remark. Since the fixed point algebra M̂^{Φ} coincides with the center Z(M̂) of M̂, (3) of Lemma 2.2 implies that P_{μ} preserves Z(M̂) ≅ ℓ^{∞}(Ĝ). Restriction of P_{μ} to Z(Ĝ) is nothing but the Markov operator on the fusion algebra Ĝ considered in [16]. Indeed, let 1_{s}, s ∈ Ĝ be the unit of Â(s). Then,

T_{r}(P_{s}(1_{t})) = \frac{1}{d(s)}(T_{r} ∗ T_{s})(1_{t}) = N_{r,s}^{t} \frac{d(t)}{d(s)},

which shows

P_{s}(1_{t}) = ∑_{r∈Ĝ} N_{r,s}^{t} \frac{d(t)}{d(s)d(r)}1_{r}.  (3.1)

(4) of the above theorem implies that the Poisson boundary of P_{μ}|_{Z(M̂)} can be realized by R^{α_{R}} = Z(N^{α}). Therefore, using T. Hayashi’s result [14, Corollary 3.5], we get the following corollary. Note that though Hayashi assumed μ to be symmetric in [14], his proof goes without any change in the general generating measure case.

Corollary 3.7 Assume that the fusion algebra Ĝ is commutative. Then with the same assumption as in Theorem 3.6, the Poisson boundary of the restriction of P_{μ} to Z(M̂) is trivial. In consequence, N^{α} is a factor.

Let us show some examples.

Examples. (1) When M is commutative, there exists a compact Lie group G such that M = L^{∞}(G). In this case, the usual asymptotic abeliance argument [30] shows that R is trivial. Therefore, H^{∞}(M̂, P_{μ}) is always trivial.

(2) When M is cocommutative, there exists a discrete group Γ such that M̂ is isomorphic to ℓ^{∞}(Γ). In this case, H^{∞}(M̂, P_{μ}) = H^{∞}(Γ, μ) as in Subsection 2.5. The above remark implies R = Z(M^{α}). The algebra M^{α} often appears as the core algebra of some subfactor coming from a Γ action [6] [7] [34] [35].

(3) The most interesting class of examples would be those non-trivial compact quantum groups with the commutative fusion algebras Ĝ, such as SU_{q}(N). The above corollary shows that in this case N^{α} is a factor and α_{R} gives rise to an ergodic action: namely, the fixed point subalgebra of R under α_{R} is trivial. In next section, we work on SU_{q}(2) and determine the structure of this ergodic action.

Before closing this section, we show a technical lemma, which will be used for the Poisson integral formula for { \atop SU_{q}(2)}. We define a unital normal completely positive map Θ : M ⟶ M̂ by

Θ(x) = (id ⊗ h) · Φ̂(x), x ∈ M.

Note that Θ is trivial if M is either commutative or cocommutative, or more generally if M is a Kac algebra.

Lemma 3.8 Let μ be a probability measure on Ĝ. Then,

(1) Every element in the image of Θ is P_{μ}-harmonic.

(2) Θ commutes with both of the right (M, δ) action and the left (M̂, δ̂)

actions, that is,

(Θ ⊗ id) · δ = Φ · Θ,
(id ⊗ Θ) · Φ̂ = δ̂· Θ.

Proof. (1) It suffices to show Q_{s} · Θ = d(s)Θ for all s ∈ Ĝ. For x ∈ M, we have

Q_{s} · Θ(x) = (id ⊗ T_{s} ⊗ h) · (δ̂⊗ id) · Φ̂(x)
= (id ⊗ T_{s} ⊗ h) · (id ⊗ Φ̂) · Φ̂(x).

We claim that for all s ∈ Ĝ the following holds:

(T_{s} ⊗ h) · Φ̂(x) = d(s)h(x), x ∈ M,

which would prove the statement. Indeed, using T_{s}(e(s)_{jk}) = f_{1}(w(s)_{k,j}), we get

(T_{s} ⊗ h) · Φ̂(x) = ∑_{j,k,l∈I_{s}} T_{s}(e(s)_{jk})h(w(s)_{l,j}^{∗}xw(s)_{l,k})
= ∑_{j,k,l∈I_{s}} f_{1}(w(s)_{k,j})h(σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗}x).

Therefore, using the property (F6) of the Woronowicz characters, the righthand side equals

∑_{j,k,l,p,q∈I_{s}} f_{1}(w(s)_{k,j})f_{−1}(w(s)_{l,p})f_{−1}(w(s)_{q,k})h(w(s)_{p,q}w(s)_{l,j}^{∗}x),

= ∑_{j,l,p,q∈I_{s}}(f_{−1} ∗ f_{1})(w(s)_{q,j})f_{−1}(w(s)_{l,p}))h(w(s)_{p,q}w(s)_{l,j}^{∗}x),
= ∑_{j,l,p∈I_{s}} f_{−1}(w(s)_{l,p}))h(w(s)_{p,j}w(s)_{l,j}^{∗}x),
= ∑_{l∈I_{s}} f_{−1}(w(s)_{ll}))h(x).

This means that (T_{s} ⊗ h) · Φ̂ is proportional to h, which proves the claim.

(2) Using the pentagon equation and invariance of h, we get the first equation:

(Θ ⊗ id) · δ(x) = (id ⊗ h ⊗ id)(V_{12}^{∗}V_{23}(1 ⊗ x ⊗ 1)V_{23}^{∗}V_{12})
= (id ⊗ h ⊗ id)(V_{13}V_{23}V_{12}^{∗}(1 ⊗ x ⊗ 1)V_{12}V_{23}^{∗}V_{13}^{∗})
= V (id ⊗ h ⊗ id) · (id ⊗ δ) · Φ̂(x)V^{∗}
= V ((id ⊗ h) · Φ̂(x) ⊗ 1)V^{∗}
= Φ · Θ(x).

Since Φ̂ is a left action of (M̂, δ̂), we have

(id ⊗ Θ) · Φ̂(x) = (id ⊗ id ⊗ h) · (id ⊗ Φ̂) · Φ̂(x)
= (id ⊗ id ⊗ h) · (δ̂⊗ id) · Φ̂(x)
= δ̂· Θ(x).

Q.E.D.

Corollary 3.9 The image of Θ consists of only scalars if and only if M is a Kac algebra. In consequence, if M is not a Kac algebra, the relative commutant R is not trivial.

Proof. It is well known that a compact Woronowicz algebra is a Kac algebra if and only if the Haar measure is a trace [2], or equivalently the Woronowicz character f_{z} does not depends on z, which is further equivalent to κ^{2} = id. (2.1.3) implies that for x ∈ M we have

Θ(x) = ∑_{s∈Ĝ} ∑_{j,k,l∈I_{s}} h(w(s)_{l,j}^{∗}xw(s)_{l,k})e(s)_{jk}
= ∑_{s∈Ĝ} ∑_{j,k,l∈I_{s}} h(σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗}x)e(s)_{jk}.

Therefore, if M is a Kac algebra we get

Θ(x) = ∑_{s∈Ĝ} ∑_{j,k,l∈I_{s}} h(w(s)_{l,k}w(s)_{l,j}^{∗}x)e(s)_{jk}
= ∑_{s∈Ĝ} ∑_{j,k,l∈I_{s}} h(κ(w(s)_{j,l}w(s)_{k,l}^{∗})x)e(s)_{jk}
= h(x)1.

Next, we assume that the image of Θ contains only scalars. Then, there exists a ∈ A such that for every s ∈ Ĝ we have

∑_{l∈Ĝ} σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗} = δ_{j,k}a.

The s = 0 case shows a = 1. Thus, this implies

κ^{2}(w(s)_{l,k}) = ∑_{j,l∈I_{s}} σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗}κ^{2}(w(s)_{l,j})
= ∑_{j,l∈I_{s}} σ_{i}^{h}(w(s)_{l,k})κ(w(s)_{j,l}^{∗}w(s)_{j,l})
= σ_{i}^{h}(w(s)_{l,k}).

The properties (F5) and (F6) of the Woronowicz characters show that this is equivalent to f_{1} = f_{−1}, which is further equivalent to ρ = ρ^{−1}. Since ρ is a positive self-adjoint operator, we get ρ = 1, which implies that M is a Kac algebra. Q.E.D.

4 Preliminaries for the SU_{q}(2) Case

In the rest of this paper, we assume that G is the quantum group SU_{q}(2) introduced by Woronowicz in [45]. For simplicity we assume 0 < q < 1, though with a minor modification our argument is also valid for −1 < q < 0. We follow [25] for the notation, and in particular, the choice of the generators of SU_{q}(2) for our exposition. Therefore some formulae we will present here appear different from those in [46] and [32] .

4-1. C(SU_{q}(2)). The “function algebra A = C(SU_{q}(2)) over SU_{q}(2) ” is the universal C^{∗}-algebra generated by four generators x, u, v, and y with the following relations:

ux = qxu, vx = qxv, yu = quy, yv = qvy,
uv = vu, xy − q^{−1}uv = yx − quv = 1,
x^{∗} = y, u^{∗} = −q^{−1}v.

For a non-negative half integer s, we set

I_{s} = {−s, −s + 1, ⋯, s}.

Then, the fundamental corepresentation {w(1/2)_{i,j}}_{i,j∈I_{1/2}} is given by

w(1/2) = ( \begin{matrix}& −1/2 & 1/2 \\ −1/2 & x & u \\ 1/2 & v & y \end{matrix} )

The coproduct δ is determined by

δ(w(1/2)_{i,j}) = ∑_{k∈I_{1/2}} w(1/2)_{i,k} ⊗ w(1/2)_{k,j}, i, j ∈ I_{1/2}.

As in the classical case, Ĝ is identified with the set of non-negative half integers, and s ∈ Ĝ is called spin. We denote by w(s)_{i,j} the unitary corepresentation of spin s obtained in [25, Theorem 2.8].

Explicit expression of two kinds of inner products of A with respective to the Haar measure h are obtained in ([25, (3.30), (3.31)]):

h(w(s)_{i,j}^{∗}w(t)_{k,l}) = δ_{s,t}δ_{i,k}δ_{j,l}q^{2(s+i)} \frac{1 − q^{2}}{1 − q^{2(2s+1)}} ,  (4.1.1)

h(w(t)_{k,l}w(s)_{i,j}^{∗}) = δ_{s,t}δ_{i,k}δ_{j,l}q^{2(s−j)} \frac{1 − q^{2}}{1 − q^{2(2s+1)}} ,  (4.1.2)

The Woronowicz characters {f_{z}} and the the modular automorphism group {σ_{t}^{h}}_{t∈R} are given by

f_{z}(w(s)_{k,l}) = δ_{k,l}q^{−2kz},  (4.1.3)
σ_{t}^{h}(w(s)_{k,l}) = q^{−2i(k+l)t}w(s)_{k,l},  (4.1.4)

For later use, we present explicit formulae of w(s)_{i,s} and w(s)_{i,−s}. As in [25], we use the following symbols: for two numbers a, q and a natural number l,

(a; q)_{0} := 1, (a; q)_{l} := ∏_{i=1}^{l−1}(1 − aq^{i}).

For two non-negative integers m ≤ n,

[ \begin{matrix}n \\ m \end{matrix} ]_{q} := \frac{(q; q)_{n}}{(q; q)_{m}(q; q)_{n−m}} .

Then, w(s)_{i,s} and w(s)_{i,−s} are given by ([25, Theorem 1.8])

w(s)_{i,s} = [ \begin{matrix}2s \\ s + i \end{matrix} ]_{q^{2}}^{1/2} u^{s−i}y^{s+i}, w(s)_{i,−s} = [ \begin{matrix}2s \\ s + i \end{matrix} ]_{q^{2}}^{1/2} x^{s−i}v^{s+i}. (4.1.5)

4-2. U_{q}(su(2)). The quantum universal enveloping algebra U_{q}(su(2)) is the ∗-algebra generated by k, k^{−1}, e, and f with the following relations [10][19]:

kk^{−1} = k^{−1}k = 1,

kek^{−1} = qe, kfk^{−1} = q^{−1}f,

ef − fe = \frac{k^{2} − k^{−2}}{q − q^{−1}} ,

k^{∗} = k, k^{−1}^{∗} = k^{−1}, e^{∗} = f.

U_{q}(su(2)) is regarded as a subalgebra of Â with the following relations:

k^{±}(w(s)_{i,j}) = δ_{i,j}q^{∓j},
e(w(s)_{i,j}) = δ_{j,i+1}\sqrt{(s + j)_{q}(s − j + 1)_{q}},
f(w(s)_{i,j}) = δ_{j,i−1}\sqrt{(s − j)_{q}(s + j + 1)_{q}}.  (4.2.1)

where for a half integer n, n_{q} stands for

n_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}} .

Note that the Woronowicz character f_{1} is given by k^{2}, and so,

ρ = R(k^{2}).

Therefore, the quantum dimension d(s) for s ∈ Ĝ is given by

d(s) = q^{−2s} + q^{−2s+2} + ⋯ + q^{2s} = (2s + 1)_{q}.

The coproduct δ̂_{0} and the antipode κ̂ on the generators are as follows:

δ̂_{0}(k^{±}) = k^{±} ⊗ k^{±},

δ̂_{0}(e) = e ⊗ k + k^{−1} ⊗ e, δ̂_{0}(f) = f ⊗ k + k^{−1} ⊗ f,
κ̂(k^{±}) = k^{∓},
κ̂(e) = −qe, κ̂(f) = −q^{−1}f.

For s ∈ Ĝ, the spin s irreducible representation π_{s} of U_{q}(su(2)) on ℓ^{2}(I_{s}) with respect to the canonical basis {ξ_{j}^{s}}_{j∈I_{s}} is given by

π_{s}(k)ξ_{j}^{s} = q^{−j}ξ_{j}^{s},
π_{s}(e)ξ_{j}^{s} = \sqrt{(s + j)_{q}(s − j + 1)_{q}}ξ_{j−1}^{s},
π_{s}(f)ξ_{j}^{s} = \sqrt{(s − j)_{q}(s + j + 1)_{q}}ξ_{j+1}^{s}.  (4.2.2)

For a left U_{q}(su(2))-module L, ξ ∈ L \ {0} is called a highest weight vector of spin s ∈ Ĝ, if the following holds:

e · ξ = 0, k · ξ = q^{s}ξ.

If ξ is such a vector, U_{q}(su(2)) · ξ is equivalent to the spin s module. The Casimir element C is a central element of U_{q}(su(2)) defined by

C = (\frac{q^{1/2}k − q^{−1/2}k^{−1}}{q − q^{−1}} )^{2} + fe,

which satisfies

π_{s}(C) = (s + \frac{1}{2})_{q}^{2}1_{s}, s ∈ Ĝ.

4-3. The Adjoint Representation. For X ∈ U_{q}(su(2)), the coproduct δ̂_{0}(X), which is a priori defined as an element of (A ⊗ A)^{∗}, actually belongs to the algebraic tensor product U_{q}(su(2)) ⊗ U_{q}(su(2)), as it is the case for the generators. Therefore, δ̂_{0}(X) has the form ∑_{i} X_{i}^{(1)} ⊗ X_{i}^{(2)}. Using this fact, we introduce the adjoint representation ad(X) of U_{q}(su(2)) on Â by

ad(X)Y := ∑_{i} X_{i}^{(1)}Y κ̂(X_{i}^{(2)}), Y ∈ Â.

It is easy to show that ad(X) is well defined and gives a representation of U_{q}(su(2)). For the generators, ad(X) is given by

ad(k)Y = kY k^{−1},
ad(e)Y = eY k^{−1} − qk^{−1}Y e, ad(f)Y = fY k^{−1} − q^{−1}k^{−1}Y f.

Lemma 4.1 With the above notation, the following hold:

(1) Let n be a non-negative integer. Then, e^{n}k^{n} is a highest weight vector of spin n with respect to the adjoint representation of U_{q}(su(2)). For s ∈ Ĝ with 2s ≥ n, π_{s}(e^{n}k^{n}) is not zero.

(2) There exists a U_{q}(su(2))-module map from ℓ^{2}(I_{1}) to U_{q}(su(2)) that sends ξ_{−1}^{1}, ξ_{0}^{1}, and ξ_{1}^{1} to

χ_{−1} = ek, χ_{0} = \frac{fe − q^{−2}ef}{\sqrt{2_{q}}} , χ_{1} = −q^{−1}kf.

Proof. (1) follows from the definition of the highest weight vectors and the above formulae. (2) follows from (4.2.2) and

\frac{1}{\sqrt{(2)_{q}}} ad(f)ek = \frac{fe − q^{−2}ef}{\sqrt{2_{q}}} ,

\frac{1}{\sqrt{(2)_{q}}} ad(f)(\frac{fe − q^{−2}ef}{\sqrt{2_{q}}}) = −q^{−1}kf.

Q.E.D.

Let Φ be the right (M, δ) action on M̂ introduced in Section 2. Thanks to (2.1.3), we have

Φ(Â(s)) ⊂ Â(s) ⊗ W (s)W (s)^{∗}, s ∈ Ĝ.

We denote by Φ_{s} the restriction of Φ to Â(s), which is given by

Φ_{s}(a) = ∑_{i,j,l,m∈I_{s}} e(s)_{ij}ae(s)_{ml} ⊗ w(s)_{i,j}w(s)_{lm}^{∗}, a ∈ Â(s).

Every finite dimensional corepresentation of A is self-contragredient, and the product W (s)W (s) decomposed into

W (0) ⊕ W (1) ⊕ ⋯ ⊕ W (2s).

Therefore, as a right (A, δ)-comodule (or equivalently left U_{q}(su(2))-module), Â(s) is decomposed into the direct sum of the spin n comodules (or the spin n modules) n = 0, 1, ⋯, 2s with multiplicity one.

As stated in Section 2, the algebra ˜ M̂ of the elements affiliated with M̂ can be identified with ∏_{s∈Ĝ} Â(s). Therefore we can extend the definition of

Φ to ˜ M̂ by setting

Φ(a) = (Φ_{s}(a_{s})) ∈ ∏_{s∈Ĝ} Â(s) ⊗ W (s)W (s)^{∗}, a = (a_{s}) ∈ ∏_{s∈Ĝ} Â(s).

For X ∈ Â, (id ⊗ X)(Φ(a)) makes sense as an element of ˜ M̂.

Lemma 4.2 Let X ∈ U_{q}(su(2)) and Y ∈ Â. Then, we have

(id ⊗ X)(Φ(R(Y))) = R(ad(X)Y).

Proof. For s ∈ Ĝ, the Â(s)-component of the left-hand side is

∑_{i,j∈I_{s}} Y (w(s)_{i,j})(id ⊗ X)(Φ(e(s)_{ij}))

= ∑_{i,j,l,m∈I_{s}} X(w(s)_{l,i}w(s)_{m,j}^{∗})Y (w(s)_{i,j})e(s)_{lm}
= ∑_{i,j,l,m∈I_{s}}(id ⊗ κ̂)(δ̂_{0}(X))(w(s)_{l,i} ⊗ w(s)_{j,m})Y (w(s)_{i,j})e(s)_{lm}
= ∑_{i,j,l,m∈I_{s}} ∑_{p}(X_{p}^{(1)} ⊗ Y ⊗ κ̂(X_{p}^{(2)}))(w(s)_{l,i} ⊗ w(s)_{i,j} ⊗ w(s)_{j,m})e(s)_{lm}
= ∑_{l,m,∈I_{s}}(ad(X)Y)(w(s)_{l,m})e(s)_{lm},

which is the Â(s)-component of the right-hand side. Q.E.D.

Let ad_{s} be the representation of U_{q}(su(2)) on Â(s) corresponding to Φ_{s} for s ∈ Ĝ; namely ad_{s}(X)a = (id ⊗ X)(Φ_{s}(a)) for a ∈ Â(s). The above two lemmas imply

Corollary 4.3 Let E_{s}, s ∈ Ĝ be a natural projection from ˜ M̂ onto Â(s). Then, for an integer n with 0 ≤ n ≤ 2s, E_{s}(R(e^{n}k^{n})) is a unique (up to scalar multiple) spin n highest weight vector in Â(s) with respect to ad_{s}.

4-4. Quantum Spheres. Let D be a C^{∗}-algebra with a right action β of (A, δ). For each s ∈ Ĝ, we set

D(s) = {x ∈ D; β(x) ∈ D ⊗ W (s)}.

In [32, Theorem 1], P. Podles completely classified those (D, β) satisfying the following three properties: (1) D(0) = C, (2) the multiplicity of D(1) is one, and (3) D(1) generates D. Such (D, β) is called a quantum sphere.

Assume that (D, β) satisfies the above three conditions and take a basis {X_{i}}_{i∈I_{1}} of D(1) satisfying

β(X_{i}) = ∑_{j∈I_{1}} X_{j} ⊗ w(1)_{j,i}.

Note that {X_{i}}_{i∈I_{1}} are generators of D by assumption. Then, either of the following occurs up to scalar multiple of the generators [32] (Our expression of the matrix w(1) is different from that in [32], and the following formulae are taken from [27]): Case I.

q^{−1}X_{0}X_{−1} − qX_{−1}X_{0} = q^{−1}X_{1}X_{0} − qX_{0}X_{1}
= (q^{−1} − q)X_{0}^{2} + X_{1}X_{−1} − X_{−1}X_{1}
= 0.

D is denoted by C(S_{q,∞}^{2}) in this case. Case II. There exists a real number c in the set

J_{q} = \{−\frac{1}{(q^{n} + q^{−n})^{2}} ; n = 2, 3, ⋯ \} ∪ [0, ∞),

such that

X_{0}^{∗} = X_{0}, X_{−1}^{∗} = −qX_{1}, X_{1}^{∗} = −q^{−1}X_{−1},
q^{−1}X_{0}X_{−1} − qX_{−1}X_{0} = (q^{−1} − q)X_{−1},
(q^{−1} − q)X_{0}^{2} + X_{1}X_{−1} − X_{−1}X_{1} = (q^{−1} − q)X_{0},
q^{−1}X_{1}X_{0} − qX_{0}X_{1} = (q^{−1} − q)X_{1},
X_{0}^{2} − qX_{−1}X_{1} − q^{−1}X_{1}X_{−1} = 1 + (q + q^{−1})^{2}c.

D is denoted by C(S_{q,c}^{2}) in this case.

The quantum spheres are completely classified by the parameter c, (in the first case, c should be regarded as ∞), and for every value c ∈ J_{q} ∪ {∞}, C(S_{q,c}^{2}) actually exists. Since C(S_{q,c}^{2})(0) = C, there exists a unique β-invariant state on C(S_{q,c}^{2}), which will be also denoted by h and called the Haar measure. The weak closure of C(S_{q,c}^{2}) in the GNS representation of h will be denoted by L^{∞}(S_{q,c}^{2}).

Among other quantum spheres, there is a distinguished one coming from a “quantum homogeneous space”. Since A is a universal algebra with the generators and relations described at the beginning of this section, there exists a homomorphism γ from A to C(T) defined by

γ(x) = z, γ(y) = \overline{z}, γ(u) = γ(v) = 0,

where z is the coordinate function of T. γ is actually a Hopf algebra homomorphism and is considered to be the restriction map to the “subgroup T of SU_{q}(2)”. It is easy to show that γ on A is given by

γ(w(s)_{i,j}) = δ_{i,j}z^{−2i}.

We set

C(T\SU_{q}(2)) = {x ∈ A; (γ ⊗ id) · δ(x) = 1 ⊗ x}.

Then, C(T\SU_{q}(2)) is a C^{∗}-subalgebra of A, which is the closure of the linear span of

⋃_{s∈Ĝ,j∈I_{s}}{w(s)_{0,j}}.

(C(T\SU_{q}(2)), δ_{|C(T\SU_{q}(2))}) is a quantum sphere with c = 0, which can be directly verified with the choice of generators

X_{−1} = w(1)_{0,−1} = \sqrt{1 + q^{2}}xv, X_{1} = w(1)_{0,1} = \sqrt{1 + q^{2}}uy,
X_{0} = w(1)_{0,0} = 1 + (q + q^{−1})uv.

The weak closure of C(T\SU_{q}(2)) in the GNS representation of h will be denoted by L^{∞}(T\SU_{q}(2)), which can be regarded as the von Neumann subalgebra of M.

Let Φ̂ be the left (M̂, δ̂) action on M defined in Subsection 2-4. The following lemma shows that Φ̂ preserves C(T\SU_{q}(2)):

Lemma 4.4 Let M(Â ⊗ C(T\SU_{q}(2))) be the multiplier algebra of Â ⊗ C(T\SU_{q}(2)). Then,

Φ̂(C(T\SU_{q}(2))) ⊂ M(Â ⊗ C(T\SU_{q}(2))).

In consequence, the restriction of Φ̂ to L^{∞}(T\SU_{q}(2)) is a left action of (M̂, δ̂) on L^{∞}(T\SU_{q}(2)).

Proof. Let a ∈ C(T\SU_{q}(2))). Then thanks to (2.1.3), Φ̂(a) has the form

∑_{s∈Ĝ,i,j∈I_{s}} e(s)_{ij} ⊗ a_{ij},

where a_{ij} = ∑_{l∈I_{s}} w(s)_{li}^{∗}aw(s)_{l,j}. To prove the statement, it suffices to show a_{ij} ∈ Â(s) ⊗ C(T\SU_{q}(2)). Indeed,

(γ ⊗ id) · δ(a_{ij})
= ∑_{l,∈I_{s}}(γ ⊗ id) · δ(w(s)_{li}^{∗}aw(s)_{l,j})
= ∑_{l,m,p,∈I_{s}}(γ ⊗ id)((w(s)_{l,p}^{∗} ⊗ w(s)_{pi}^{∗})δ(a)(w(s)_{lm} ⊗ w(s)_{m,j}))
= ∑_{l,m,p,∈I_{s}} γ(w(s)_{l,p}^{∗}γ(w(s)_{lm}) ⊗ w(s)_{pi}^{∗}aw(s)_{m,j}
= ∑_{l,p,∈I_{s}} 1 ⊗ w(s)_{pi}^{∗}aw(s)_{p,j}
= 1 ⊗ a_{ij}.

Q.E.D. Remark. As one can see in the proof, the above lemma actually holds for an arbitrary pair of a compact quantum group and its compact quantum subgroup.

4-5. Some equalities. We end this section with a few equalities involving the q-binomial coefficients, which will be used in next section. The following are elementary and easily shown by induction:

Lemma 4.5 Let n be a non-negative integer. Then, (1) For an indeterminate z,

(z; q)_{n} = ∑_{j=0}^{n} q^{j(j−1)/2} [ \begin{matrix}n \\ j \end{matrix} ]_{q} (−z)^{j}.

(2) For two elements a, b in an algebra satisfying ba = qab,

(a + b)^{n} = ∑_{j=0}^{m} [ \begin{matrix}n \\ j \end{matrix} ]_{q} a^{j}b^{n−j}.

The next lemma is less elementary than the previous one though it is still easy to show.

Lemma 4.6 The following equalities hold:

(1) For non-negative integers m, n, and p with 0 ≤ p ≤ m + n,

[ \begin{matrix}m + n \\ p \end{matrix} ]_{q} = ∑_{j} q^{j(j+n−p)} [ \begin{matrix}m \\ j \end{matrix} ]_{q} [ \begin{matrix}n \\ p − j \end{matrix} ]_{q} ,

where j runs over all integers satisfying

max{0, p − n} ≤ j ≤ min{m, p}.

(2) For an indeterminate z and an non-negative integer n,

\frac{(q^{n+1}; q)_{n}}{(z; q)_{n+1}} = ∑_{j=0}^{n} q^{j(2n+1)} \frac{(q^{−n}; q)_{j}(q^{−n}z; q)_{j}}{(q; q)_{j}(z; q)_{j+1}} .

Proof. (1) Take two elements a, b in an algebra satisfying ba = qab. Lemma 4.6 implies

(a + b)^{m+n} = (a + b)^{m}(a + b)^{n}
= ∑_{j=0}^{m} ∑_{i=0}^{n} [ \begin{matrix}m \\ j \end{matrix} ]_{q} [ \begin{matrix}n \\ i \end{matrix} ]_{q} a^{m−j}b^{j}a^{i}b^{n−i}

= ∑_{j=0}^{m} ∑_{i=0}^{n} [ \begin{matrix}m \\ j \end{matrix} ]_{q} [ \begin{matrix}n \\ i \end{matrix} ]_{q} q^{ij}a^{m−j+i}b^{n−i+j},

which shows the statement.

(2) We introduce a polynomial g(z) of degree at most n by

g(z) = ∑_{j=0}^{n}(−1)^{j}q^{j(j+2n+1)/2} [ \begin{matrix}n \\ j \end{matrix} ]_{q} (q^{−n}z; q)_{j}(q^{n}z; q^{−1})_{n−j}.

Then, to prove the statement, it suffices to show g(z) = (q^{n+1}; q)_{n} for n + 1 distinct z. Setting z = q^{n−m} with 0 ≤ m ≤ n, we get

g(q^{n−m}) = ∑_{j=0}^{m} q^{j(j+n−m)} [ \begin{matrix}n \\ j \end{matrix} ]_{q} \frac{(q; q)_{2n−m}(q; q)_{m}}{(q; q)_{n+j−m}(q; q)_{m−j}}

= (q; q)_{n} ∑_{j=0}^{m} q^{j(j+n−m)} [ \begin{matrix}m \\ j \end{matrix} ]_{q} [ \begin{matrix}2n − m \\ n − j \end{matrix} ]_{q} .

(1) implies that this is equal to

(q; q)_{n} [ \begin{matrix}2n \\ n \end{matrix} ]_{q} = \frac{(q; q)_{2n}}{(q; q)_{n}} = (q^{n+1}; q)_{n}.

Q.E.D.

5 SU_{q}(2), Spin 1/2 Case

The goal of the rest of this paper is to apply the general theory established in Section 3 to a concrete example; we determine the structure of the relative commutant R, or equivalently, the structure of H^{∞}(M̂, P_{μ}) for G = SU_{q}(2). In this section, we work on the spin 1/2 Markov operator P_{1/2}. Based on the analysis of eigenvalues of P_{1/2} in this section, we treat the general case in next section. For simplicity, P_{1/2} and Q_{1/2} will be denoted by P and Q respectively.

As we have already seen in (4) of Theorem 3.6, H^{∞}(M̂, P) is the weak closure of the linear span of the spectral subspaces H^{∞}(M̂, P)(s), s ∈ Ĝ. So, we first determine the structure of H^{∞}(M̂, P)(s) for each s. For this purpose, it is enough to obtain the highest weight vectors because the fact

Φ · P = (P ⊗ id) · Φ

implies that P commutes with the adjoint representation of U_{q}(su(2)). Thanks to Corollary 4.3, we already know all the integer spin highest weight vectors in each Â(t), t ∈ Ĝ for the adjoint representation. (Note that Φ has only integer spins and it is actually an action of SO_{q}(3) rather than SU_{q}(2).) Therefore, our main concern is how to assemble them to make bounded P -harmonic elements.

From now on, we omit the symbol R when there is no possibility of

confusion, and identify Â, ˜ M̂, and ∏_{s∈Ĝ} Â(s). For every X ∈ Â and q-trace T_{s}, (id⊗T_{s})(δ̂(X)) makes sense as an element of Â. Therefore, we can extend P and Q to Â in a natural way. The extensions will be denoted by P̃ and Q̃.

Lemma 5.1 Let n be a non-negative integer and e, k, C ∈ U_{q}(su(2)) be as in the previous section. Then,

(1)  Q̃(e^{n}k^{n}) = (q^{n} + q^{−n})e^{n}k^{n},

(2)  Q̃(Ce^{n}k^{n}) = (q^{n+2} + q^{−(n+2)})Ce^{n}k^{n} + 2(n + \frac{3}{2})_{q}(\frac{1}{2})_{q}e^{n}k^{n}.

Proof. (1) Thanks to Lemma 4.5, we have

δ̂(e^{n}k^{n}) = (e ⊗ k + k^{−1} ⊗ e)^{n}(k^{n} ⊗ k^{n})
= ∑_{j=0}^{n} [ \begin{matrix}n \\ j \end{matrix} ]_{q^{2}} (k^{−j}e^{n−j}k^{n} ⊗ e^{j}k^{2n−j})

= ∑_{j=0}^{n} [ \begin{matrix}n \\ j \end{matrix} ]_{q^{2}} q^{j(j−n)}(e^{n−j}k^{n−j} ⊗ e^{j}k^{2n−j}).

Since T_{1/2}(e^{j}k^{2n−j}) ≠ 0 only if j = 0, we get the formula.

(2) Direct computation shows

δ̂(C) = C ⊗ k^{2} + k^{−2} ⊗ C
− \frac{(q + q^{−1})(k^{−2} ⊗ k^{2}) + 2 − 2(1 ⊗ k^{2} + k^{−2} ⊗ 1)}{(q − q^{−1})^{2}}
+ k^{−1}e ⊗ fk + fk^{−1} ⊗ ke.

Thus, the only terms of δ̂(Ce^{n}k^{n}) that may survive in (id ⊗ T_{1/2}) are the following:

Ce^{n}k^{n} ⊗ k^{2(n+1)} − \frac{2}{(q − q^{−1})^{2}} e^{n}k^{n} ⊗ (k^{2n} − k^{2n+2})

+ q^{−2n}e^{n}k^{n−2} ⊗ (q^{n+1}n_{q}fe + C + \frac{2 − (q + q^{−1})k^{2}}{(q − q^{−1})^{2}} )k^{2n}.

In fact, the last term does not survive because Q̃(Ce^{n}k^{n}) should be a highest weight vector of spin n for the adjoint representation (it can be shown by direct computation as well). Thus, we obtain the statement. Q.E.D.

Now we determine the action of P on the highest weight vectors.

Lemma 5.2 Let E_{s} be the natural projection from Â onto Â(s), s ∈ Ĝ. For a natural number l, we set x_{n,l} = E_{\frac{l−1}{2}} (e^{n}k^{n}), ( Note that x_{n,l} ≠ 0, if and only if l ≥ n + 1 ). Then,

P (x_{n,n+1}) = \frac{1}{(q + q^{−1})(n + 2)_{q}} x_{n,n+2},

P (x_{n,l}) = \frac{(l − n)_{q}}{(q + q^{−1})(l + 1)_{q}} x_{n,l+1} + \frac{(l + n)_{q}}{(q + q^{−1})(l − 1)_{q}} x_{n,l−1}, l ≥ n + 2.

Proof. As mentioned in the remark after Theorem 3.6, we have

Q(Â(s)) ⊂ Â(s − \frac{1}{2}) ⊕ Â(s + \frac{1}{2}).

Since Q(x_{n,l}) is again a highest weight vector of spin n for the adjoint representation, there exist two numbers a_{n}(l+1, l), l ≥ n and a_{n}(l−1, l), l ≥ n+1 with a_{n}(n, n + 1) = 0, a_{n}(n + 1, n) = 0 satisfying

Q(x_{n,l}) = a_{n}(l + 1, l)x_{n,l+1} + a_{n}(l − 1, l)x_{n,l−1}, l ≥ n + 1.

We claim that the following hold:

E_{\frac{l−1}{2}} (Q̃(e^{n}k^{n})) = (a_{n}(l, l − 1) + a_{n}(l, l + 1))x_{n,l}, l ≥ n + 1,

E_{\frac{l−1}{2}} (Q̃(Ce^{n}k^{n})) = [(\frac{l − 1}{2})_{q}^{2}a_{n}(l, l − 1) + (\frac{l + 1}{2})_{q}^{2}a_{n}(l, l + 1)]x_{n,l}, l ≥ n + 1.

Indeed, we set

ξ = (q + q^{−1})Λ_{h}(qx + q^{−1}y) ∈ H_{h}.

Then, (4.1.1) implies that for every a ∈ Â,

T_{1/2}(a) = ⟨aξ, ξ⟩.

Thus, we get

⟨Q̃(e^{n}k^{n})Λ_{h}(w(s)_{0,i}), Λ_{h}(w(s)_{0,j})⟩
= ⟨V^{∗}(1 ⊗ R(e^{n}k^{n}))V (Λ_{h}(w(s)_{0,i}) ⊗ ξ), Λ_{h}(w(s)_{0,j}) ⊗ ξ⟩
= ∑_{p,r∈I_{s}}⟨Λ_{h}(w(s)_{0,p}) ⊗ R(e^{n}k^{n})w(s)_{p,i}ξ, (Λ_{h}(w(s)_{0,r}) ⊗ w(s)_{r,j}ξ)⟩.

Since w(s)_{p,i}ξ ∈ Λ_{h}(W (s + \frac{1}{2})) + Λ_{h}(W (s − \frac{1}{2})), we have

R(e^{n}k^{n})w(s)_{p,i}ξ = (E_{s+ \frac{1}{2}} (e^{n}k^{n}) + E_{s− \frac{1}{2}} (e^{n}k^{n}))w(s)_{p,i}ξ.

Thus, we can replace e^{n}k^{n} with x_{n,2s} + x_{2s+2} in the above equation, and so

E_{s}(Q̃(e^{n}k^{n})) = E_{s}(Q(x_{n,2s}) + Q(x_{n,2s+2})).

This proves the first equality of the claim, and the second one can be shown in the same way.

Now using Lemma 5.1, we get

a_{n}(l, l − 1) + a_{n}(l, l + 1) = q^{n+2} + q^{−(n+2)}, l ≥ n + 1,

(\frac{l − 1}{2})_{q}^{2}a_{n}(l, l − 1) + (\frac{l + 1}{2})_{q}^{2}a_{n}(l, l + 1)

= (q^{n+2} + q^{−(n+2)})(\frac{l}{2})_{q}^{2} + 2(n + \frac{3}{2})_{q}(\frac{1}{2})_{q}, l ≥ n + 1.

Solving these, we get

a_{n}(l, l − 1) = \frac{(l − n − 1)_{q}}{l_{q}} , a_{n}(l, l + 1) = \frac{(l + n + 1)_{q}}{l_{q}} .

Q.E.D.

Thanks to Corollary 4.3, a ∈ Â is a highest weight vector of spin n if and only if there exists a sequence {a_{l}}_{l=1}^{∞} such that a_{l} = 0 for l ≤ n and

a = (a_{2s+1}x_{n,2s+1}) ∈ ∏_{s∈Ĝ} Â(s).

Moreover, the above lemma implies that a is an eigenvector of P̃ for an eigenvalue λ if and only if {a_{l}} satisfies the following recurrence relation:

λ(q + q^{−1})(n + 1)_{q}a_{n+1} = (2(n + 1))_{q}a_{n+2},
λ(q + q^{−1})l_{q}a_{l} = (l + n + 1)_{q}a_{l+1} + (l − n − 1)_{q}a_{l−1}, l ≥ n + 2. (5.1)

Therefore, for each eigenvalue λ, the eigenspace of P̃ in Â contains each integer spin spectral subspace for the adjoint representation with multiplicity exactly one. In this sense, the eigenvalue problem of P̃ is completely solved. To settle the problem for P, namely to single out bounded eigenvectors from the above ones, we need some estimates.

It is easy to show the following lemma from (4.2.2):

Lemma 5.3 For each non-negative integer n, there exist two positive constant C_{1} and C_{2}, depending only on q and n, such that

C_{1}q^{−nl} ≤ ||x_{n,l}|| ≤ C_{2}q^{−nl}, l ≥ n + 1.

For a non-negative integer n and a non-zero complex number α with |α| ≤ 1, we introduce a sequence {a_{l}^{n,α}}_{l=n+1}^{∞} determined by the following recurrence relation:

a_{n+1}^{n,α} = 1, a_{n+2}^{n,α} = \frac{(1 + α^{2})(1 − q^{2(n+1)})}{1 − q^{4(n+1)}} = \frac{1 + α^{2}}{1 + q^{2(n+1)}} ,

(α + α^{−1})(1 − q^{2l})a_{l}^{n,α}
= α^{−1}(1 − q^{2(l+n+1)})a_{l+1}^{n,α} + α(1 − q^{2(l−n−1)})a_{l−1}^{n,α}, l ≥ n + 2.

Lemma 5.4 Let {a_{l}^{n,α}}_{l=n+1}^{∞} be as above. Then,

(1) For α^{2} ≠ 1, {a_{l}^{n,α}}_{l=n+1}^{∞} is bounded. Moreover, if |α| = 1, we have

lim_{l→∞}(a_{l}^{n,α} − \frac{α^{2l}(q^{2(n+1)}; q^{2})_{n+1}}{α^{2(n+1)}(α^{−2}; q^{2})_{n+1}} ) = \frac{(q^{2(n+1)}; q^{2})_{n+1}}{(α^{2}; q^{2})_{n+1}} .

If |α| < 1, we have

lim_{l→∞} a_{l}^{n,α} = \frac{(q^{2(n+1)}, q^{2})_{n+1}}{(α^{2}, q^{2})_{n+1}} .

In particular, for α = q,

lim_{l→∞} a_{l}^{n,q} = [ \begin{matrix}2n + 1 \\ n \end{matrix} ]_{q^{2}} .

(2) For α^{2} = 1,

lim_{l→∞} \frac{a_{l}^{n,α}}{l} = \frac{(q^{2}; q^{2})_{2n+1}}{(q^{2}; q^{2})_{n}^{2}} .

(3) If α is a positive number, then a_{l}^{n,α} > 0 for every l ≥ n + 1.

Proof. (1) It is easy to show that for fixed n and α, there exists c > 0 such that |a_{l}^{n,α}| ≤ c^{l} for all l ≥ n + 1. Thus, we can introduce an analytic function g_{n,α}(z) defined on a neighborhood of 0 by

g_{n,α}(z) = ∑_{l=n+1}^{∞} a_{l}^{n,α}z^{l}.

The recurrence relation of {a_{l}^{n,α}}_{l=n+1}^{∞} implies the following function equation:

(1−z)(1−α^{2}z)g_{n,α}(z) = q^{2n}(1−q^{−2n}z)(1−q^{−2n}α^{2}z)g_{n,α}(q^{2}z)+(1−q^{2(2n+1)})z^{n+1}.

Therefore, g_{n,α} has analytic continuation to

D = C \ ({q^{−2j}; j ∈ Z_{≥0}} ∪ {α^{−2}q^{−2j}; j ∈ Z_{≥0}}).

The above function equation implies

(1 − q^{2(2n+1)})z^{n+1}q^{2l(2n+1)} \frac{(q^{−2n}z; q^{2})_{l}(α^{2}q^{−2n}z; q^{2})_{l}}{(z; q^{2})_{l+1}(α^{2}z; q^{2})_{l+1}}

= \frac{(q^{−2n}z; q^{2})_{l}(α^{2}q^{−2n}z; q^{2})_{l}}{(z; q^{2})_{l}(α^{2}z; q^{2})_{l}} q^{2nl}g_{n,α}(q^{2l}z)
− \frac{(q^{−2n}z; q^{2})_{l+1}(α^{2}q^{−2n}z; q^{2})_{l+1}}{(z; q^{2})_{l+1}(α^{2}z; q^{2})_{l+1}} q^{2n(l+1)}g_{n,α}(q^{2(l+1)}z).

It is easy to show that

\frac{(q^{−2n}z; q^{2})_{l}(α^{2}q^{−2n}z; q^{2})_{l}}{(z; q^{2})_{l}(α^{2}z; q^{2})_{l}} q^{2nl}g_{n,α}(q^{2l}z)

uniformly converges to 0 on every compact subset of D when l tends to infinity. Therefore, we get

g_{n,α}(z) = (1 − q^{2(2n+1)})z^{n+1} ∑_{l=0}^{∞} q^{2l(2n+1)} \frac{(q^{−2n}z; q^{2})_{l}(α^{2}q^{−2n}z; q^{2})_{l}}{(z; q^{2})_{l+1}(α^{2}z; q^{2})_{l+1}} .

If α^{2} ≠ 1 and |α| = 1, z = 1 and z = α^{−2} are poles of order at most one. Indeed using Lemma 4.6, we get

lim_{z→1}(1 − z)g_{n,α}(z) = (1 − q^{2(2n+1)}) ∑_{l=0}^{n} q^{2l(2n+1)} \frac{(q^{−2n}; q^{2})_{l}(α^{2}q^{−2n}; q^{2})_{l}}{(q^{2}; q^{2})_{l}(α^{2}; q^{2})_{l+1}}

= \frac{(q^{2(n+1)}; q^{2})_{n+1}}{(α^{2}; q^{2})_{n+1}} ,

lim_{z→α^{−2}}(1 − α^{2}z)g_{n,α}(z) = \frac{(1 − q^{2(2n+1)})}{α^{2(n+1)}} ∑_{l=0}^{n} q^{2l(2n+1)} \frac{(q^{−2n}; q^{2})_{l}(α^{−2}q^{−2n}; q^{2})_{l}}{(q^{2}; q^{2})_{l}(α^{−2}; q^{2})_{l+1}}

= \frac{(q^{2(n+1)}; q^{2})_{n+1}}{α^{2(n+1)}(α^{−2}; q^{2})_{n+1}} .

Therefore, there exists an analytic function g_{n,α}^{1}(z) on |z| < q^{−2} such that

g_{n,α}^{1}(z) = g_{n,α}(z) − \frac{(q^{2(n+1)}; q^{2})_{n+1}}{(α^{2}; q^{2})_{n+1}} \frac{1}{1 − z} − \frac{(q^{2(n+1)}; q^{2})_{n+1}}{α^{2(n+1)}(α^{−2}; q^{2})_{n+1}} \frac{1}{1 − α^{2}z}
= ∑_{l=0}^{∞}(a_{l}^{n,α} − \frac{(q^{2(n+1)}; q^{2})_{n+1}}{(α^{2}; q^{2})_{n+1}} − \frac{α^{2l}(q^{2(n+1)}; q^{2})_{n+1}}{α^{2(n+1)}(α^{−2}; q^{2})_{n+1}} )z^{l},

where we set a_{l}^{n,α} = 0 for l ≤ n. This shows the statement for |α| = 1. The |α| < 1 case can be treated in a similar way.

(2) In a similar way as above, we get the following for α^{2} = 1:

lim_{l→∞}(1 − z)^{2}g_{n,α}(z) = (1 − q^{2(2n+1)}) ∑_{l=0}^{n} q^{2l(2n+1)} \frac{(q^{−2n}; q^{2})_{l}^{2}}{(q^{2}; q^{2})_{l}^{2}}

= \frac{(q^{2}; q^{2})_{2n+1}}{(q^{2}; q^{2})_{n}^{2}}

which shows the statement.

(3) Assume that α is positive number less than or equal to 1. Let r be a positive number. Then, the recurrence relation is equivalent to

α^{−1}(1 − q^{2(l+n+1)})(a_{l+1}^{n,α} − ra_{l}^{n,α}) = αr^{−1}(1 − q^{2(l−n−1)})(a_{l}^{n,α} − ra_{l−1}^{n,α}) + b_{l}a_{l}^{n,α},

b_{l} := α(1 − r^{−1}) + α^{−1}(1 − r) − q^{2l}(α + α^{−1} − α^{−1}rq^{2(n+1)} − αr^{−1}q^{−2(n+1)}),

where l ≥ n+1 and a_{n}^{n,α} is understood as 0. Therefore, to prove the statement by induction, it suffices to show that there exists r such that b_{l} > 0 for all l ≥ n + 1, which is equivalent to the following two:

α(1 − r^{−1}) + α^{−1}(1 − r) = \frac{(1 − r)(r − α^{2})}{rα} ≥ 0.

b_{n+1} = \frac{(1 − q^{4(n+1)})}{α}(\frac{1 + α^{2}}{1 + q^{2(n+1)}} − r) > 0.

Such a choice of r is always possible. Q.E.D.

For a non-negative integer n and a complex number λ, we introduce a sequence {b_{l}^{n,λ}}_{l=1}^{∞} with b_{l}^{n,λ}=0 for l ≤ n satisfying the following recurrence relation (See [26] for a similar recurrence relation appearing in the spherical functions on the quantum spheres):

b_{n+1}^{n,λ} = 1, b_{n+2}^{n,λ} = \frac{λ(q + q^{−1})}{1 + q^{2(n+1)}},

λ(q + q^{−1})(1 − q^{2l})b_{l}^{n,λ} = (1 − q^{2(l+n+1)})b_{l+1}^{n,λ} + (1 − q^{2(l−n−1)})b_{l−1}^{n,λ}, l ≥ n + 1.

Note that if α is a complex number with |α| ≤ 1 satisfying

α + α^{−1} = λ(q + q^{−1}),

then, b_{l}^{n,λ} = α^{n+1−l}a_{l}^{n,α}. Moreover, a_{l} = q^{(n+1)l}b_{l}^{n,λ} satisfies (5.1). Thus, we can construct eigenvectors {h^{λ}(n)_{j}}_{j∈I_{n}} of P̃ for λ by setting

h^{λ}(n)_{−n} = (q^{2s(n+1)}b_{2s+1}^{n,λ}x_{n,2s+1}) ∈ ∏_{s∈Ĝ} Â(s),

h^{λ}(n)_{j} = \frac{1}{\sqrt{(n + 1 − j)_{q}(n + j)_{q}}} ad(f)h^{λ}(n)_{j−1}.

Note that Lemma 5.3 and Lemma 5.4 imply that h^{λ}(n)_{j} is bounded if and only if q ≤ |α| ≤ 1. For λ = 1, h^{λ}(n)_{j} will be simply denoted by h(n)_{j}. Summing up the results we have obtained so far, we get the following theorem:

Theorem 5.5 For a non-negative integer n and a complex number λ, we denote by M̂(P, λ)(n) and Â(P̃, λ)(n) the eigenspaces of P and P̃ for the eigenvalue λ with spin n for the adjoint representation. Then,

(1) {h^{λ}(n)_{j}}_{j∈I_{n}} is a basis of Â(P̃, λ)(n).

(2) M̂(P, λ)(n) = Â(P̃, λ)(n), namely h^{λ}(n)_{j} is bounded for j ∈ I_{n} if and only if λ = ξ + iη, ξ, η ∈ R satisfies

ξ^{2} + (\frac{1 + q^{2}}{1 − q^{2}} )^{2}η^{2} ≤ 1.

As a corollary, we get

Corollary 5.6 Let n be a non-negative integer. Then,

(1) The multiplicity of spectral subspace H^{∞}(M̂, P)(n) for the adjoint representation is exactly one, and {h(n)_{j}}_{j∈I_{n}} is a basis of H^{∞}(M̂, P)(n).

(2) A complex number λ is an eigenvalue of P restricted to the spin n spectral subspace of M̂ if and only if it is an eigenvalue of P restricted to the center Z(M̂).

(3) P̃ restricted to the center Z(Â) has a positive eigenvector for eigenvalue λ if and only if

λ ≥ \frac{2}{q + q^{−1}} .

Now we decide the algebra structure of R using the Choi-Effros product on H^{∞}(M̂, P). Let χ_{i}, i = −1, 0, 1 be as in Lemma 4.1. Then, we have

h(1)_{j} = (a_{2s+1}^{1,q}q^{2s+1}E_{s}(χ_{j})) ∈ ∏_{s∈Ĝ} Â(s),

h(n)_{−n} = (a_{2s+1}^{n,q}q^{(2s+1)n}x_{n,2s+1}) ∈ ∏_{s∈Ĝ} Â(s).

It is straightforward to show the following:

Lemma 5.7 Let C be the Casimir element. Then,

χ_{0}^{∗} = χ_{0}, χ_{−1}^{∗} = −qχ_{1}, χ_{1}^{∗} = −q^{−1}χ_{1},

q^{−1}χ_{0}χ_{−1} − qχ_{−1}χ_{0} = \frac{1 − q^{−2}}{\sqrt{2_{q}}} (C + \frac{2}{(q − q^{−1})^{2}} )(q^{−1} − q)χ_{−1},

χ_{0}^{2} − qχ_{−1}χ_{1} − q^{−1}χ_{1}χ_{−1} = \frac{(1 − q^{−2})^{2}}{2_{q}} [(C + \frac{2}{(q − q^{−1})^{2}} )^{2} − \frac{(q + q^{−1})^{2}}{(q − q^{−1})^{4}} ].

Thanks to (3) of Lemma 5.4, for each non-negative integer n we introduce a family of nearest neighbor transition probability on the set N_{≥n+1} of natural numbers larger than n by

p_{n}(l, l + 1) = \frac{(1 − q^{2(l+n+1)})a_{l+1}^{1,q}}{(1 + q^{2})(1 − q^{2l})a_{l}^{1,q}} , l ≥ n + 1,

p_{n}(l, l − 1) = \frac{q^{2}(1 − q^{2(l−n−1)})a_{l−1}^{1,q}}{(1 + q^{2})(1 − q^{2l})a_{l}^{1,q}} , l ≥ n + 2.

Since

p(l, l + 1) ∼ \frac{1}{1 + q^{2}} > \frac{1}{2} (l → ∞),

these random walks are transient. Lemma 5.2 implies

P (q^{nl}a_{l}^{n,q}x_{n,l})

= p_{n}(l + 1, l)q^{n(l+1)}a_{l+1}^{n,q}x_{n,l+1} + p_{n}(l − 1, l)q^{n(l−1)}a_{l−1}^{n,q}x_{n,l−1}, l ≥ n + 1,

where p_{n}(n, n + 1) should be understood as 0. We denote by p_{n}^{(m)}(j, l) the m-step transition probability from j to l.

Lemma 5.8 With the above notation, we have

(1)  h(1)_{0}^{∗} = h(1)_{0}, h(1)_{−1}^{∗} = −qh(1)_{1}, h(1)_{1}^{∗} = −q^{−1}h(1)_{1},

(2)

s− lim_{m→∞} P^{m}(q^{−1}h(1)_{0}h(1)_{−1} −qh(1)_{−1}h(1)_{0}) = \frac{−(1 − q^{6})}{(1 − q^{2})^{2}\sqrt{2_{q}}} (q^{−1} −q)h(1)_{−1}.

(3) s− lim_{m→∞} P^{m}(h(1)_{0}^{2} −qh(1)_{−1}h(1)_{1} −q^{−1}h(1)_{1}h(1)_{−1}) = [\frac{(1 − q^{6})}{(1 − q^{2})^{2}\sqrt{2_{q}}}]^{2}.

(4)  s − lim_{m→∞} P^{m}(h(1)_{−1}^{n}) = \frac{q^{n(n−1)/2}(1 − q^{6})^{n}}{(1 − q^{2})^{n} [ \begin{matrix}2n + 1 \\ n \end{matrix} ]_{q^{2}}} h(n)_{−n}.

Proof. (1) is an immediate consequence of the previous Lemma 5.7. (2) For l ≥ n + 1, we set

ξ(l) = −\frac{q^{−1}(1 + q^{2l})a_{l}^{1,q}}{\sqrt{2}_{q}} .

Then, Lemma 5.7 implies

q^{−1}h(1)_{0}h(1)_{−1} − qh(1)_{−1}h(1)_{0} = (ξ(2s + 1)a_{2s+1}^{1,q}q^{2s+1}E_{s}(χ_{−1})) ∈ ∏_{s∈Ĝ} Â(s).

Therefore we get

P^{m}(q^{−1}h(1)_{0}h(−1) − qh(1)_{−1}h(1)_{0})
= (ξ^{(m)}(2s + 1)a_{2s+1}^{1,q}q^{2s+1}E_{s}(χ_{−1})) ∈ ∏_{s∈Ĝ} Â(s),

where

ξ^{(m)}(l) = ∑_{j=n+1}^{∞} p_{1}^{(m)}(l, j)ξ(j).

As the random walk determined by p_{1}(l, j) is transient, (which is in fact more than enough), for fixed l and a finite set F we have

lim_{m→∞} ∑_{j∈F} p_{1}^{(m)}(l, j) = 0.

Since ξ(l) has limit as l goes to infinity, we get

lim_{m→∞} ξ^{(m)}(l) = lim_{j→∞} ξ(j) = −\frac{q^{−1}}{\sqrt{2_{q}}} [ \begin{matrix}3 \\ 1 \end{matrix} ]_{q^{2}} = −\frac{1 − q^{6}}{q(1 − q^{2})\sqrt{2_{q}}} ,

where we use Lemma 5.4. Therefore, we get the statement.

(3) In a similar way, we set

η(l) = \frac{[(1 + q^{2l})^{2} − (q + q^{−1})q^{2l}](a_{l}^{1,q})^{2}}{2_{q}(1 − q^{2})^{2}} .

Then,

h(1)_{0}^{2} − qh(1)_{−1}h(1)_{1} − q^{−1}h(1)_{1}h(1)_{−1} = (η(2s + 1)1_{s}).

(Note that a_{l}^{0,q} = 1). The same argument using p_{0}(l, j) implies the statement from

lim_{j→∞} η(j) = \frac{(1 − q^{6})^{2}}{(1 − q^{2})^{4}2_{q}} .

(4) We set

ζ(l) = q^{n(n−1)/2} \frac{(a_{l}^{1,q})^{n}}{a_{l}^{n,q}} .

Then,

h(1)_{−1}^{n} = (ζ(2s + 1)a_{2s+1}^{n,q}q^{n(2s+1)}x_{n,2s+1}).

The same argument using p_{n}(l, j) implies the statement from the limit

lim_{j→∞} ζ(j) = \frac{q^{n(n−1)/2}(1 − q^{6})^{n}}{(1 − q^{2})^{n} [ \begin{matrix}2n + 1 \\ n \end{matrix} ]_{q^{2}}} .

Q.E.D.

Using Podles’s classification of quantum spheres [32], we get the main result of this section.

Theorem 5.9 With the above notation, the following hold:

(1) There exists a ∗-isomorphism θ_{1} from the relative commutant R onto L^{∞}(T\SU_{q}(2)) such that

φ_{R} = h · θ_{1},
(θ_{1} ⊗ id) · α_{R} = δ · θ_{1}.

(2) There exists a completely positive isometry θ_{0} from H^{∞}(M̂, P) onto L^{∞}(T\SU_{q}(2)) commuting with the (M, δ) actions such that for every non-negative integer n and j ∈ I_{n}, we have

θ_{0}(h(n)_{j}) = \frac{(−1)^{n}q^{n/2}(1 − q^{2(2n+1)})}{(1 − q^{2})^{n}(1 − q^{2(n+1)})} [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2} w(n)_{0,j}.

Proof. (1) Let θ be the map constructed in Theorem 3.6. We set

X_{i} = −\frac{q^{−1/2}(1 + q^{2})^{1/2}(1 − q^{2})^{2}}{(1 − q^{6})} θ(h(1)_{i}), i = −1, 0, 1.

Since θ commutes with the (M, δ) actions, {X_{i}}_{i∈I_{1}} is a basis of the spectral subspace R(1) satisfying

α_{R}(X_{i}) = ∑_{j∈I_{1}} X_{j} ⊗ w(1)_{j,i}.

Let D be the C^{∗}-algebra generated by {X_{i}}_{i∈I_{1}}. Then, (D, α_{|D}) is isomorphic to one of the quantum spheres. Lemma 5.8 implies

X_{0}^{∗} = X_{0}, X_{−1}^{∗} = −qX_{1}, X_{1}^{∗} = −q^{−1}X_{−1},
q^{−1}X_{0}X_{−1} − qX_{−1}X_{0} = (q^{−1} − q)X_{−1},
X_{0}^{2} − qX_{−1}X_{1} − q^{−1}X_{1}X_{−1} = 1,

which shows that (D, α_{|D}) is isomorphic to C(T\SU_{q}(2)). In particular, D contains each integer spin spectral subspace with multiplicity one. Since the multiplicity of R(s) is the same as that of H^{∞}(M̂, P)(s) for every s ∈ Ĝ, Corollary 5.7 implies that R(s) has multiplicity one for every integer s and 0 for every half-integer s. Thus, R(n) ⊂ D for every non-negative integer n. Lemma 3.3 implies that D is weakly dense in R. Since φ_{R} is a faithful normal state on R that is invariant under α_{R}, the isomorphism θ_{1} from D to C(T\SU_{q}(2)) defined by

θ_{1}(X_{i}) = w(1)_{0,i}, i = −1, 0, 1,

extends to a ∗-isomorphism from R onto L^{∞}(T\SU_{q}(2)) satisfying the desired properties.

(2) We set θ_{0} = θ_{1} · θ, which is a completely positive isometry from H^{∞}(M̂, P) to L^{∞}(T\SU_{q}(2)). Using (4.1.5), we get

w(n)_{0,−n} = [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2} x^{n}v^{n} = \frac{q^{−n(n−1)/2}}{(1 + q^{2})^{n/2}} [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2} (w(1)_{0,1})^{n}.

Thus,

θ_{0}(h(1)_{−1})^{n} = \frac{(−1)^{n}q^{n/2}(1 − q^{6})^{n}}{(1 − q^{2})^{2n}(1 + q^{2})^{n/2}} (w(1)_{0,−1})^{n}
= \frac{(−1)^{n}q^{n^{2}/2}(1 − q^{6})^{n}}{(1 − q^{2})^{2n} [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2}} w(n)_{0,−n}.

On the other hand, (2) of Theorem 3.6 and (4) of Lemma 5.8 imply

θ_{0}(h(1)_{−1})^{n} = \frac{q^{n(n−1)/2}(1 − q^{6})^{n}}{(1 − q^{2})^{n} [ \begin{matrix}2n + 1 \\ n \end{matrix} ]_{q^{2}}} θ_{0}(h(n)_{−n}).

Thus, we get

θ_{0}(h(n)_{−n}) = \frac{(−1)^{n}q^{n/2} [ \begin{matrix}2n + 1 \\ n \end{matrix} ]_{q^{2}}}{(1 − q^{2})^{n} [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2}} w(n)_{0,−n}

= \frac{(−1)^{n}q^{n/2}(1 − q^{2(2n+1)})}{(1 − q^{2})^{n}(1 − q^{2(n+1)})} [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2} w(n)_{0,−n}.

The general statement for {h(n)_{j}}_{j∈I_{n}} follows from the fact that θ_{0} commutes with the (M, δ) actions. Q.E.D.

In view of the classical theory of the Poisson boundaries, for a harmonic element a ∈ H^{∞}(M̂, P), θ_{0}(a) is considered to be the “boundary value” of a on T\SU_{q}(2). The following theorem provides a way to recover a from its “boundary value” using integration by the Haar measure.

Theorem 5.10 (Poisson Integral) Let Θ be the map from M to H^{∞}(M̂, P) introduced in Section 3, that is,

Θ(a) = (id ⊗ h) · Φ̂(a), a ∈ M,

and let Θ_{0} be restriction of Θ to L^{∞}(T\SU_{q}(2)). Then, Θ_{0} is the inverse of θ_{0}. In particular, Θ_{0} is a complete positive isometry from L^{∞}(T\SU_{q}(2)) onto H^{∞}(M̂, P) commuting both of the right (M, δ) actions and the left (M̂, δ̂) actions. ( Note that the (M̂, δ̂) action on L^{∞}(T\SU_{q}(2)) is well defined thanks to Lemma 4.4.)

Proof. Since θ_{0} = θ_{1} · θ is a normal map, so is Θ_{0} · θ_{0}. Thus, to prove the statement it suffices to show Θ_{0} · θ_{0}(h(n)_{−n}) = h(n)_{−n} for each non-negative integer n because Θ_{0} · θ_{0} commutes with the right (M, δ) actions and the linear span of {h(n)_{j}}_{n,j} is dense in weak topology. Since Θ_{0} · θ(h(n)_{−n}) is a highest weight vector of spin n, Corollary 5.6 implies that Θ_{0} · θ(h(n)_{−n}) is proportional to h(n)_{−n}. Therefore, we compare the e(n/2)_{−n/2,n/2} components of h(n)_{−n} and Θ_{0} · θ(h(n)_{−n}) to show that they are equal. The former is obtained from (4.2.1), and it is

\frac{q^{3n/2}(q^{2}; q^{2})_{n}}{(1 − q^{2})^{n}} .

To obtain the latter, first we compute the e(n/2)_{−n/2,n/2} component of Θ_{0}(w(n)_{0,−n}), which is, by definition, given by

∑_{j∈I_{n/2}} h(w(n/2)_{j,−n/2}^{∗}w(n)_{0,−n}w(n/2)_{j,n/2})

= ∑_{j∈I_{n/2}} h(σ_{i}^{h}(w(n/2)_{j,n/2})w(n/2)_{j,−n/2}^{∗}w(n)_{0,−n})
= h([ ∑_{j∈I_{n/2}} w(n/2)_{j,−n/2}σ_{i}^{h}(w(n/2)_{j,n/2})^{∗}]^{∗}w(n)_{0,−n}).

Using (4.1.4), (4.1.5), and Lemma 4.5, we get

∑_{j∈I_{n/2}} w(n/2)_{j,−n/2}σ_{i}^{h}(w(n/2)_{j,n/2})^{∗}

= ∑_{l=0}^{n} q^{2l} [ \begin{matrix}n \\ l \end{matrix} ]_{q^{2}} x^{n−l}v^{l}(u^{n−l}y^{l})^{∗}

= (−q^{−1})^{n} ∑_{l=0}^{n} q^{l(l−1)}(−q^{4})^{l} [ \begin{matrix}n \\ l \end{matrix} ]_{q^{2}} x^{n}v^{n}
= (−q^{−1})^{n}(q^{4}; q^{2})_{n}x^{n}v^{n}
= \frac{(−1)^{n}(q^{2}; q^{2})_{n+1}}{q^{n}(1 − q^{2}) [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2}} w(n)_{0,−n}.

This shows that the e(n/2)_{−n/2,n/2} component of Θ_{0}(w(n)_{0,−n}) is

\frac{(−1)^{n}(q^{2}; q^{2})_{n+1}}{q^{n}(1 − q^{2}) [ \begin{matrix}2n \\ n \end{matrix} ]_{q^{2}}^{1/2}} h(w(n)_{0,−n}^{∗}w(n)_{0,−n}).

Thus, (4.1.1) and Theorem 5.9 imply that the e(n/2)_{−n/2,n/2} component of Θ_{0} · θ(h(n)_{−n}) is

\frac{(q^{2}; q^{2})_{n}(1 − q^{2(2n+1)})}{q^{n/2}(1 − q^{2})^{n+1}}h(w(n)_{0,−n}^{∗}w(n)_{0,−n}) = \frac{q^{3n/2}(q^{2}; q^{2})_{n}}{(1 − q^{2})^{n}} .

Q.E.D.

6 SU_{q}(2), General Case

Now, we proceed to the general case; namely let μ ≠ δ_{0} be a finitely supported probability measure on Ĝ. The goal of this section is to show that the “Poisson boundary” for P_{μ} does not depends on the choice of μ and it is actually L^{∞}(T\SU_{q}(2)).

Let {U_{n}(z)}_{n=0}^{∞} be the Tchebichef polynomials of second kind, which are defined by the recurrence relation

U_{0}(z) = 1, U_{1}(z) = 2z,
U_{n+1}(z) = 2zU_{n}(z) − U_{n−1}(z), n ≥ 1,

or equivalently defined by

U_{n}(cosh t) = \frac{sinh(n + 1)t}{sinh t}.

Comparing the above recurrence relation with the fusion rules of Ĝ, we get

Q_{s} = U_{2s}(\frac{Q}{2}),

d(s) = (2s + 1)_{q} = U_{2s}(\frac{q + q^{−1}}{2} ).

Lemma 6.1 Let U′_{n}(z) be the derivative of U_{n}(z). Then,

U′_{n}(\frac{q + q^{−1}}{2} ) > 0, n ≥ 1.

Proof. This follows from

U_{n+2}(cosh t)′ − U_{n}(cosh t)′ = 2(n + 2) sinh(n + 2)t, n ≥ 0.

Q.E.D.

In the SU_{q}(2) case, μ is a generating measure if and only if the support of μ contains a half-integer spin. Therefore, if μ is supported only by integer spins, the action α should be treated as an SO_{q}(3) action rather than SU_{q}(2). We first treat the generating case.

Theorem 6.2 Let μ be a finitely supported generating measure on Ĝ. Then,

H^{∞}(M̂, P_{μ}) = H^{∞}(M̂, P).

In consequence, there exits an isomorphism from R of the corresponding ITP action onto L^{∞}(T\SU_{q}(2)) commuting with the (M, δ) actions.

Proof. We define a polynomial f(z) by

f(z) = ∑_{s∈Ĝ} \frac{μ(s)U_{2s}(\frac{(q+q^{−1})z}{2})}{d(s)}.

Then, we have P_{μ} = f(P) and f(1) = 1. Thus, we get

H^{∞}(M̂, P_{μ}) ⊃ H^{∞}(M̂, P).

Thanks to Lemma 6.1, there exists a polynomial g(z) such that

f(z) − 1 = (z − 1)g(z), g(1) ≠ 0.

This implies that for every non-negative integer n and a P_{μ}-harmonic element a of spin n,

g(P)(a) ∈ H^{∞}(M̂, P)(n).

Since g(1) ≠ 0, there exist b ∈ H^{∞}(M̂, P)(n) such that g(P)(a) = g(P)(b). We set a_{1} = a−b, which satisfies g(P)(a_{1}) = 0. Suppose a_{1} ≠ 0. Then, there would exist 1 ≠ λ ∈ C such that g(λ) = 0 and λ would be an eigenvalue of P restricted to the spectral subspace of M̂ of spin n. However, this implies, thanks to (2) of Corollary 5.6, that h^{λ}(0) is bounded and

P_{μ}(h^{λ}(0)) − h^{λ}(0) = (P − I)(g(P)(h^{λ}(0))) = 0,

that is, h^{λ}(0) is a non-trivial harmonic element of P_{μ} restricted to Z(M̂). This contradicts Corollary 3.7, and so we get a = b ∈ H^{∞}(M̂, P). Since the Choi-Effros product is uniquely determine by the operator system structure of H^{∞}(M̂, P), we get the result of R from that of the spin 1/2 case. Q.E.D.

Let M̂_{I} be the von Neumann algebra direct sum of Â(n) over all nonnegative integers n. Then, M̂_{I} is regarded as the dual Hopf von Neumann algebra of SO_{q}(3). We denote by E_{I} the projection from M̂ onto M̂_{I}. When μ is supported only by integer spins, P_{μ,I} stands for the restriction of P_{μ} to M̂_{I}. We set

H^{∞}(M̂, P)_{I} = {E_{I}(a) ∈ M̂_{I}; a ∈ H^{∞}(M̂, P)},

which actually coincides with H^{∞}(M̂_{I}, P_{1,I}).

Theorem 6.3 Let μ ≠ δ_{0} be a finitely supported measure on Ĝ whose support contains integer spins only. Then, we have

H^{∞}(M̂_{I}, P_{μ,I}) = H^{∞}(M̂, P)_{I}.

Moreover, there exits an isomorphism from R of the corresponding ITP action onto L^{∞}(T\SU_{q}(2)) commuting with the (M, δ) actions.

Proof. In the same way as above, we introduce a polynomial f(z) by

f(z) = ∑_{s∈Ĝ} \frac{μ(s)U_{2s}(\frac{(q+q^{−1})z}{2})}{d(s)}.

Note that f(z) is a polynomial of z^{2} because it is the case for U_{n}(z) for even n. As before, we have P_{μ} = f(P) and f(1) = 1, and so

H^{∞}(M̂_{I}, P_{μ,I}) ⊃ H^{∞}(M̂, P)_{I}.

f(1) = 1 implies that there exists a polynomial g(z) such that

f(z) − 1 = (z^{2} − 1)g(z^{2}), g(1) ≠ 0.

This implies that for every non-negative integer n and a P_{μ,I}-harmonic element a of spin n,

P^{2}(g(P^{2})(a)) = g(P^{2})(a).

Thus, g(P^{2})(a) + P (g(P^{2})(a)) is P -harmonic. As before, there exists b ∈ H^{∞}(M̂, P)(n) such that

g(P^{2})(a) + P (g(P^{2})(a)) = g(P^{2})(b).

We set a_{1} = a − E_{I}(b), which satisfies g(P^{2})(a_{1}) = 0. Suppose a_{1} ≠ 0. Then, there would exist 1 ≠ λ ∈ C such that g(λ) = 0 and λ would be an eigenvalue of P^{2} restricted to the spectral subspace of M̂_{I} of spin n for the adjoint representation. Let a_{2} be the corresponding eigenvector. We claim λ ≠ 0. Indeed, if g(0) = 0, we would have f(0) = 1. However, for n ≠ 0

|\frac{U_{2n}(0)}{d(n)}| < 1.

Since μ is a probability measure that is not equal to δ_{0}, we get

|f(0)| < 1,

which shows the claim. We set

a_{3} = a_{2} + \frac{P (a_{2})}{\sqrt{λ}}.

Then, a_{3} is an eigenvector of P for the eigenvalue \sqrt{λ} such that a_{3} has spin n for the adjoint representation. Thus, (2) of Corollary 5.6 implies that h^{\sqrt{λ}}(0) is bounded and

P_{μ}(E_{I}(h^{\sqrt{λ}}(0))) − E_{I}(h^{\sqrt{λ}}(0)) = (P^{2} − I)(g(P^{2})(E_{I}(h^{\sqrt{λ}}(0))) = 0,

that is, E_{I}(h^{\sqrt{λ}}(0)) is a harmonic element of P_{μ,I} restricted to Z(M̂_{I}). Therefore, Corollary 3.7 implies that E_{I}(h^{\sqrt{λ}}(0)) is a scalar. However, the definition of h^{\sqrt{λ}}(0) shows that E_{I}(h^{\sqrt{λ}}(0)) is a scalar only if \sqrt{λ} = ±1, which is contradiction. Thus, we get a = E_{I}(b) ∈ H^{∞}(M̂, P)_{I}. Since the Choi-Effros product is uniquely determine by the operator system structure of H^{∞}(M̂, P)_{I}, we can compute it using P^{2} instead of P_{μ}, and so the statement for R follows from that of the spin 1/2 case. Q.E.D.

7 Questions and Comments

We end the paper with a few questions and comments.

(1) Let λ be a positive number larger than or equal to 2/(q + q^{−1}). Then (3) of Lemma 5.4 implies that h^{λ}(0) is a positive invertible element affiliated with Z(M̂). Therefore, we can introduce the “Doob transformation P^{λ}” by setting

P^{λ}(a) = \frac{P̃(h^{λ}(0)a)}{λh^{λ}(0)}, a ∈ M̂.

P^{λ} is again a Markov operator of M̂ commuting with the (M, δ) action. a ∈ M̂ is a P^{λ}-harmonic element if and only if h^{λ}(0)a is an eigenvector of P̃ for λ. (2) of Lemma 5.4 shows that {a_{l}^{n,α}} and {a_{l}^{0,α}} have the same asymptotics, up to a non-zero scalar multiple, as l tends to infinity. Therefore for every s ∈ Ĝ, H^{∞}(M̂, P^{λ})(s) has the same multiplicity as H^{∞}(M̂, P)(s) does. This means that the weak closure of the linear span of

⋃_{s∈Ĝ} H^{∞}(M̂, P^{λ})(s),

with the Choi-Effros product gives rise to the quantum sphere L^{∞}(S_{q,c}^{2}) for some c. Note that we do not know if this operator system coincides with H^{∞}(M̂, P^{λ}) in the λ ≠ 1 case because we do not have a statement corresponding to (2) of Lemma 3.3 in general, or in other words, we do not have a natural realization of the Poisson boundaries for λ ≠ 1. The same argument as above actually implies c = 0 for all λ. It would be interesting to seek for a natural realization of the von Neumann algebra structure of H^{∞}(M̂, P^{λ}).

(2) As we have seen in (4) of Lemma 2.2 (or Lemma 4.4), there is a natural left (M̂, δ̂) action on H^{∞}(M̂, P) (or on L^{∞}(T\SU_{q}(2))). Therefore, there exists a left (M̂, δ̂) action on the relative commutant R. What does this action mean? Probably, the origin of the action will be the dual action α̂ acting on the relative commutant

(N ⋊_{α} SU_{q}(2)) ∩ N′.

(3) We go back to the general situation in Section 3. Let L be the linear span of

⋃_{s∈Ĝ,j,k∈I_{s}} ∑_{l∈I_{s}} σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗}.

Then, it is easy to show that L is a left coideal which is closed under ∗. In fact, L is a dense ∗-subalgebra of C(T\SU_{q}(2)) in the SU_{q}(2) case. Is L an algebra in general? In view of Corollary 3.9 and the SU_{q}(2) case, the weak closure of L is a candidate of the realization of H^{∞}(M̂, P_{μ}) when the fusion algebra Ĝ is commutative. We introduce an element K ∈ ∏_{s∈Ĝ}(Â(s) ⊗ L) by

K = ( ∑_{j,k,l∈I_{s}} e(s)_{jk} ⊗ σ_{i}^{h}(w(s)_{l,k})w(s)_{l,j}^{∗}) ∈ ∏_{s∈Ĝ}(Â(s) ⊗ L).

The proof of Corollary 3.9 implies that the map Θ is given by

Θ(a) = (id ⊗ h)(K(1 ⊗ a)) = (id ⊗ h)((1 ⊗ a)K^{∗}).

Therefore, at least in the SU_{q}(2) case, K deserves to be called the “Poisson kernel”. It is easy to show that K is also given by

K = (id ⊗ κ)(Φ(ρ^{−2})).

(4) It is known that if a discrete group Γ allows a generating probability measure whose Poisson boundary is trivial, Γ is necessarily amenable [22]. (The converse is not true.) Is it also true in our situation? More precisely, if H^{∞}(M̂, P_{μ}) is trivial for some generating measure, is the fusion algebra Ĝ amenable in the sense of [16]? Note that if Ĝ is amenable, M is necessarily a Kac algebra [15] (cf. Corollary 3.9.)

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