Spherical Representations of $C^*$-Flows II: Representation System and Quantum Group Setup

This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form $G<G\times G$ (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups.


Introduction
In [35] we introduced the notion of (α t , β)-spherical representations for C * -flows α t and developed its general theory including Vershik-Kerov's ergodic method and the spectral decomposition of such a representation for a certain class of C * -flows.The present paper is a sequel to that paper, and still attempts to provide some general results.In fact, we will introduce a natural algebraic structure for a certain class of C * -flows, which plays a similar rôle as dimension groups in Vershik-Kerov's theory.We will also explain how the general theory given in the previous paper and the first part of this paper nicely fits the asymptotic representation theory for quantum groups initiated by Ryosuke Sato [31].
Let G = lim − → G n be an inductive limit of compact groups, which is usually not locally compact.When G is not locally compact, we cannot utilize standard methods, based on C * -algebras, to investigate unitary representations of G. Nevertheless, Vershik-Kerov's asymptotic representation theory (see [36,37] and Kerov's thesis [12]) as well as Olshanski's spherical representation theory (see [23,24,25]) work well for both the infinite symmetric group S ∞ = lim − → S n (that is locally compact) and the infinite-dimensional unitary group U(∞) = lim − → U(n) (that is not locally compact) in an almost parallel fashion.This is a bit surprising phenomenon to us, because the natural C * -algebras associated with those groups are of different kinds.Actually, the group C * -algebra C * (S ∞ ) is an AF-algebra, while we think, through the analysis of qunatum groups, that the natural C * -algebra of U(∞) should be a C * -inductive limit of atomic W * -algebras since the associated branching graph is not locally finite.
Vershik-Kerov's theory should be understood as a theory for characters (rather than representations), and is applicable even for tracial states of general AF-algebras.In their theory, the concept of dimension groups (or ordered K 0 -groups) of AF-algebras plays an important rôle; see [38,39].In fact, the dimension group of the group C * -algebra C * (S ∞ ) is computed from the so-called Littlewood-Richardson ring of the inductive sequence S n (and hence it admits a natural ring structure).This suggests us that the part of dealing with dimension groups in Vershik-Kerov's theory is essential from the viewpoint of representation theory.However, we cannot apply the idea of dimension groups to general inductive limits of compact groups directly, because those inductive limits may not be dealt with within the class of AF-algebras since their branching graphs are no longer locally finite as pointed out above.Although Boyer [3] attempted to generalize Vershik-Kerov's use of dimension groups to U(∞), etc. based on Geissinger's idea [10], we focus on Olshanski's recent work [26] that introduced an analogous ring for U(∞) based on the theory of symmetric functions.In fact, we are interested in understanding the construction of Olshanski's ring in a general setup like Vershik-Kerov's theory [38] dealing with traces on general AF-algebras.
As mentioned above, Sato [31] initiated the study of asymptotic representation theory for quantum groups motivated by Gorin's work [11].The main idea of that work [31] is to replace tracial states with KMS states with respect to deformation (or scaling) flows that arise as the effect of q-deformation of classical groups.Thus we will construct a natural operator system (and its relative) for a certain class of flows on inductive limits of atomic W * -algebras (whose special cases are AF-algebras), which plays a rôle of dimension groups in Vershik-Kerov's works and generalizes Olshanski's ring for U(∞) naturally.Moreover, we will answer, from the viewpoint of spherical unitary representations, the questions of why KMS states and which inverse temperature are appropriate in the quantum group setting.Note that there are no special representation-theoretic reasons for those choices in Sato's approach because of the character of Vershik-Kerov's theory that his approach is modeled after.
This paper consists of two parts.The contents are as follows.
In the first part, for a given continuous, inductive flow on a locally atomic W * -algebra, we will introduce an operator system (and its relative) playing a rôle of dimension groups in Vershik-Kerov's theory, and explain that the resulting operator system has a natural pairing with all the locally normal KMS states with respect to the flow.We also introduce a certain module structure on the operator system under a certain assumption.We will also show that (the relative of) that operator system coincides with Olshanski's ring in the case of U(∞).
The second (and main) part concerns basic theory of spherical unitary representations for inductive limits G = lim − → G n of compact quantum groups.We will explain how our general theory works for those G. Namely, we will naturally define the pair G < G × G and its spherical unitary representations.Then we will show that any spherical unitary representations fall into the framework of (α t , β)-spherical representations.As a consequence, we will be able to justify the use of KMS states in the asymptotic representation theory for quantum groups and to see which inverse temperature should be selected.
We will also upgrade Sato's notion of quantized characters [31] to "quantized character functions over G".This notion is a natural generalization of functions g ∈ G → χ(g) ∈ C with characters χ on an ordinary group G.
In closing of the second part, we will explain how to apply the theory we have developed so far to the example U q (∞) from quantum unitary groups U q (n).As an application of the notion of quantized character functions, we will give a clearer interpretation of Gorin's q-Schur generating functions (see [11]) in terms of U q (∞).We try to make it clear that all the computations can essentially be done by means of quantized universal enveloping algebras U q gl(n) (although the formulation of spherical unitary representations certainly needs the operator algebraic notion of compact quantum groups due to its character of unitary representations).

Notations
We will use several kinds of tensor products throughout; ⊗, ⊗, ⊗ min and ⊗ max denote algebraic, W * -algebraic, minimal C * -algebraic and maximal C * -algebraic tensor products, respectively.The reader can find their basic facts in Brown-Ozawa's book [6].
Similarly to our previous paper [35], we encourage the reader who is not familiar with operator algebras to consult Bratteli-Robinson's two-volume book [4,5] for basic facts on operator algebras.We will also use the notion of operator systems and their relatives.The necessary basic facts on them are summarized, e.g., in [17, Section 2], and a standard reference book on them is Paulsen's famous one [28].

General setup and notations
Let A n , n ≥ 0, be an inductive sequence of atomic W * -algebras with separable preduals such that A 0 = C and that each inclusion A n → A n+1 is unital and normal, that is, A n is a unital W * -subalgebra of A n+1 .Denote by Z n all the minimal projections of the center Z(A n ).By the general structure of atomic W * -algebras, each zA n with z ∈ Z n is identified with all the bounded operators B(H z ) on a unique separable Hilbert space H z .We will write dim(z) := dim(H z ) for short.
With the inductive sequence A n is associated the following graded graph, called the Bratteli diagram: The vertex set is Z := n≥0 Z n , a disjoint union, and the edge set is E := n≥1 E n , a disjoint union, with E n := {(z 2 , z 1 ) ∈ Z n × Z n−1 ; z 2 z 1 = 0}.Each pair (z 2 , z 1 ) ∈ E n should be read as a directed edge from z 1 to z 2 so that its source s(z 2 , z 1 ) and range r(z 2 , z 1 ) are z 1 and z 2 , respectively.We also have a function m : E → N ∪ {∞}, called the multiplicity function, defined by m(z 2 , z 1 ) := dim(z 2 z 1 (A n ∩ (A n−1 ) )) 1/2 with (z 2 , z 1 ) ∈ E n .As explained in [35,Section 9.1] one can reconstruct the inductive sequence A n from the data (Z, E, m).However, the multiplicity function is not so important in the analysis below.Another important function on the edges E will be recalled in Section 3.2.
Let A = lim − → A n be an inductive (or direct) limit C * -algebra (see, e.g., [9,Chapter 2]).Assume that we have a flow α t : R A such that α t (A n ) = A n holds for every t ∈ R and n ≥ 0, and moreover, the restriction of α t to each A n , denoted by α t n : R A n , is continuous in the utopology (see, e.g., [35,Lemma 7.1] for this topology).This assumption forces every flow α t n to fix elements in Z(A n ).See [35,Lemma 7.1] for this fact.In particular, the restriction of α t to zA n with z ∈ Z n gives a continuous flow α t z on zA n = B(H z ) in the u-topology.Throughout this paper, we will consider only an inverse temperature β ∈ R, for which there is a (unique) normal (α t z , β)-KMS state τ (α t z ,β) on each zA n with z ∈ Z n .This is the case for all β ∈ R, when every zA n = B(H z ) is finite dimensional.(See, e.g., [35,Section 7].)For every a ∈ A n we write instead.)This normal conditional expectation can be characterized by the (operator-valued) KMS condition denotes the σ-weakly dense * -subalgebra of α t nanalytic elements of A n .When α t n is the trivial flow or β = 0, the conditional expectation E (α t n ,β) must be the center-valued trace.We also remark that holds if n > m ≥ 0. See [35,Lemma 7.2(3)].For the ease of notation, we often write τ z := τ (α t z ,β) and E n := E (α t n ,β) if the flow α t and the inverse temperature β are clear from context.We will denote by K ln β (α t ) the convex set of all locally normal (α t , β)-KMS states on A, which is a face of all the (α t , β)-KMS states K β (α t ).Here, we recall that an (α t , β)-KMS state ω on A is locally normal if the restriction of ω to each A n is normal.Our previous work shows that any locally normal (α t , β)-spherical representation is the standard form of A associated with a locally normal (α t , β)-KMS state on A, where we borrow the term "standard form" from theory of von Neumann (or W * -)algebras.Thus, the investigation of K ln β (α t ) is of particular importance in our study.

Representation (operator) system
We keep the setting in Section 2 and fix an inverse temperature β with the requirement there throughout.In this section, we are seeking for a suitable algebraic structure playing a rôle of dimension groups in Vershik-Kerov's asymptotic representation theory (see [38,39]).

Preparatory consideration
Every Z(A n ) is naturally identified with all the bounded complex-valued functions Accordingly, the predual Z(A n ) * is identified with all the summable complex-valued functions 1 (Z n ) over Z n in such a way that each ω ∈ Z(A n ) * is considered as a function z ∈ Z n → ω(z) ∈ C. Thus, we have a natural pairing In what follows, we will freely identify Let ω ∈ K ln β (α t ) be arbitrarily given.The restriction of ω to each zA n = B(H z ) with z ∈ Z n is a normal (α t z , β)-KMS state.By the uniqueness of KMS states, we have ω = ω(z)τ z on zA n .Hence ω n := ω Z(An) , the restriction to Remark that this equation must hold for every n.Moreover, the right-most side of (3.1) clearly gives a normal (α t n , β)-KMS state on A n .Consequently, the duality pair (Z(A n ) * , Z(A n )) contains essential information of K ln β (α t ).It is known, see, e.g., [30,Proposition 6.1], that the K 0 -group K 0 (A n ) is isomorphic, via the center-valued trace, to the pointwise additive group consisting of f ∈ ∞ (Z n , R), the real-valued bounded functions on Z n , such that f (z) ∈ {k/ dim(z); k ∈ Z} for every z ∈ Z n .Moreover, the equivalence class [1] of the unit 1 ∈ A n in K 0 (A n ) corresponds, via the isomorphism, to the constant function f ≡ 1.Consequently, we may and do assume that (K 0 (A n ), [1]) is an additive subgroup of the hermitian elements Keeping these observations in mind, we will consider (Z(A n ), 1) as an operator system (an ordered * -vector space with an additional order structure) in place of (K 0 (A n ), [1]).

Constructions
Motivated by the consideration in the previous section we will introduce an operator system S(α t , β) as an analog of dimension groups associated with (α t , β) and a kind of its predual.

Inductive and projective sequences
The diagram holds as an element in ∞ (Z n+1 ).In what follows, we simply write Θ n,m , which coincides with the restriction of E n to Z(A m ) thanks to (2.1), for each pair n > m.
We then consider the (pre)dual map Θ * n+1,n : and hence holds as an element of 1 (Z n ) = Z(A n ) * .We will write Φ n,n+1 := Θ * n+1,n in what follows.In this way, we have the inductive sequence in the category of operator systems and the projective sequence in the category of Banach spaces, that is, every Φ n,n+1 is contractive since every Θ n+1,n is UCP.We remark that inductive sequence (3.4) is precisely Banach space dual to projective sequence (3.5).
Here we recall a definition from [35,Section 7].
Note that [κ(z , z)] defines a stochastic matrix over Z n+1 × Z n for each n ≥ 0. See [35,Lemma 7.9].Identities (3.2) and (3.3) can be rewritten as respectively.The predual Z(A n ) * is equipped with a natural involution ω * (a) := ω(a * ) for a ∈ Z(A n ), and this involution is nothing but the standard involution on 1 (Z n ) taking pointwise complex-conjugation.
Observe that all the hermitian elements in ), becomes a (real) Banach lattice.By (3.7) every morphism Φ n,n+1 is positive with respect to this Banach lattice structure.Namely, we have in the category of Banach lattices (whose morphisms should be positive and contractive), and projective sequence (3.5) is obtained as the complexification of this sequence.Hence, projective sequence (3.5) can be understood in the category of complex Banach lattices.See, e.g., [29, p. 157] for the notion of complex Banach lattices.However, we will not take any viewpoint of Banach lattices in what follows.
Remark 3.2.Assume for a while that α t is the trivial flow or β = 0 and further that all A n are finite dimensional, that is, A is an AF-algebra.It is not difficult to see that the matrix Because of this remark, we will regard inductive sequence (3.4) as a counterpart of inductive sequences of dimension groups.We also point out that this is completely consistent with the picture that Olshanski explained, in [26,Section 1.4], about his view of "Fourier transform" on U(∞).

Inductive limit
The concept of inductive limits of operator systems was first introduced and used by Kirchberg [13], but he treated it in the category of norm-complete operator systems.A systematic study of the concept was recently conducted by Mawhinney and Todorov [17] in the category of general (not necessarily norm-complete) operator systems.We will follow this recent study to discuss the inductive limit of (3.4).Definition 3.3.
be the inductive limit in the category MOU of matrix ordered * -vector spaces with matrix order unit (say MOU spaces), equipped with matrix order unit ë and UCP maps Θ∞,n : Z(A n ) → S(α t , β) such that Θ∞,n+1 • Θ n+1,n = Θ∞,n holds for every n ≥ 0.
We call these S(α t , β) and S(α t , β) the representation system and the representation operator system, respectively, of the C * -flow α t at inverse temperature β.Remarks 3.4.
1.By its construction, S(α t , β) is also the inductive limit in the category OU of ordered * -vactor spaces with order unit (say OU spaces) as well as even in the category of vector spaces.Thus, for any sequence of (unital positive) linear maps Ψ n : Z(A n ) → V with a single (OU) vector space V such that Ψ n+1 • Θ n+1,n = Ψ n holds for every n ≥ 0, the correspondence Θ∞,n (f ) → Ψ n (f ) defines a well-defined (unital positive) linear map from S(α t , β) to V. See the proof of [17,Theorem 3.5].This property allows us to describe the vector space S(α t , β) itself as follows.Let n≥0 Z(A n ) be the algebraic direct sum of the Z(A n ).Define a transition operator P : With the identity operator I on n≥0 Z(A n ), the vector space S(α t , β) is identified with the quotient space n≥0 by the image of P −I, and Θ∞,n is given by the composition of the inclusion map Z(A n ) → n≥0 Z(A n ) and the quotient map.This can be shown by using [17,Remark 3.1]; actually, One may regard each of these "real subspaces" as a counterpart of "dimension group" of A to study (α t , β)-spherical representations.
The representation operator system S(α t , β) has the following special feature: Proposition 3.5.Any unital positive map Ψ : S(α t , β) → T with another operator system T is automatically CP.
Proof .We need to recall the detailed construction of operator system inductive limits in [17,Section 4.2].Write S = S(α t , β) and S := S(α t , β) for the ease of notation.We will use the standard notations such as all the k × k matrices M k ( S) whose entries are from S, and Since Z(A m ) is a commutative C * -algebra, Ψ • q S • Θ∞,m must be CP (see, e.g., [28,Theorem 3.11]).Therefore, (Ψ • q S ) (k) Θ(k) ∞,n (X) ≥ 0 so that Ψ • q S is CP.We will confirm that Ψ itself is CP.Choose an arbitrary q . By construction, for any r > 0, there are S to this identity we have . By the Archimedean property of T we conclude that Ψ (k) q (k) S (Y ) ≥ 0.

Projective limit
Since every Φ n,n+1 is contractive as remarked in Section 3.2.1 we can take the projective limit of (3.5) in the category of Banach spaces.The projective limit is Here is a possible duality relation between L(α t , β) and S(α t , β).
We will clarify what rôle the Banach space L(α t , β) plays in the next section.

Locally normal states on S(α t , β)
We begin with a definition.
Definition 3.7.An ω ∈ S(α t , β) * is said to be locally normal, if there is an The set of all locally normal states on S(α t , β) is denoted by S ln (S(α t , β)), whose topology is the weak * one.
A locally normal state on S(α t , β) is defined similarly with Θ∞,n in place of Θ ∞,n , and the set of those states is denoted by S ln ( S(α t , β)).
Theorem 3.11.The following assertions hold: ) with ν n := ν Zn , the restriction of ν to Z n , for each n is well defined, injective and affine.

S ln (S(α
There are affine homeomorphisms such that any composition of these three maps along the arrows becomes the identity map. Here, map (a) is the affine homeomorphism immediately follows from the definition of κ (α t ,β) -harmonicity and (3.7).Being injective and affine is trivial.
(2) By Lemma 3.9, any element of S ln (S(α t , β)) is of the form ι(ω) with ω ∈ L(α t , β).Then, ω) Zn for every n ≥ 0. By the definition of L(α t , β) together with (3.7) one easily sees that ν falls in H + 1 (κ (α t ,β) ) and moreover ι(ν • ) = ω by the construction of ν.Hence we have shown that ) .On the other hand, we start with an arbitrary ν ∈ H and equals 1 when f ≡ 1.Thus, ι(ν • ) must be a state on S(α t , β).Since ι(ν • ) has already been known to be locally normal by Lemma 3.9, we have ι(ν • ) ∈ S ln (S(α t , β)).Hence we have confirmed item (2). ( Thus, by the proof of [35,Proposition 7.4] the sequence (E * n (ω)) defines the desired element ). Hence we have seen that map (c) is well defined.The injectivity of map (c) is clear from the construction, since n A n is norm-dense in A. Also, it is easy to see that Assume that ω λ → ω in S ln (S(α t , β)) (with respect to the weak * topology).For any a ∈ A n we have ) (with respect to the weak * topology).Thus, map (c) is continuous.
We then prove that map (b) is continuous.Assume that ν λ → ν in H + 1 (κ (α t ,β) ) (pointwisely).It is a standard task (see the proofs of [35, Theorem 7.8 and Proposition 7.10]) to show that We finally confirm that the diagram commutes in the sense of the statement.Choose an arbitrary ω ∈ K ln β (α t ).For any a ∈ A n we have This fact together with the bijectivity of both maps (a), (b) implies that map (c) is bijective too.The commutativity also shows that maps (b), (c) are homeomorphisms.

Left/right multiplicative structure
Throughout this section, we assume that the inductive sequence A n in question has the following structure, called a left multiplicative structure: For each pair m, n ≥ 0, there is a normal injective where the horizontal arrow is the natural inclusion.
We call an inductive sequence A n with left multiplicative structure ι m,n a left multiplicative inductive sequence.The notion of right multiplicative structure is defined analogously with changing (L) into (R) For each n ≥ 0, where the horizontal arrow is the natural inclusion.
We remark that the discussion below also works for right multiplicative structures with trivial modifications.
Okada's abstract formulation of Littlewood-Richardson rings [22] motivated us to introduce this structure, but item (L) or (R) is a tricky part and formulated to hold in the quantum group setup; see Section 4.4.
We start with a technical lemma, which follows from the uniqueness of τ z = τ (α t z ,β) .
Lemma 3.12.For each triple , m, n ≥ 0 and any we have the following: Proof .These can be proved in the same way as in [35, Lemma 7.2] using the uniqueness of normal (α t z , β)-and (α t z , β)-KMS states on z A +m and z A m+n , respectively, where requirement (m3) is important.Corollary 3.13.For each triple , m, n ≥ 0 we have because all the involved maps are normal.
We have where the second equality follows from Lemma 3.12(1) with x = z ι ,m (z 1 ⊗ z 2 ), the third from (m2), and the fifth from Lemma 3.12(2) with y = z ι m,n (z 2 ⊗ z 3 ).Note that the fourth line is ). Hence we are done.
The corollary implies the next theorem.Remark that items (2), (3) below are naturally expected from item (1) and Remarks 3.4 (2).In what follows, the topology on Σ(α t , β) is the relative one induced, via the natural embedding, from the product topology on n≥0 Z(A n ) of the σ-weak topology on each Z(A n ).
Theorem 3.14.With the left multiplicative structure ι m,n we have the following: and its unit is 1 0 .Here, the unit of Z(A n ) is denoted by 1 n when it is considered as an element of Σ(α t , β).Moreover, the multiplication is separately continuous.
3. For every n ≥ 0 and Proof .
(1) Since all the involved maps are linear, it suffices to confirm the associativity and that 1 ∈ Z(A 0 ) = C is a unit.By Corollary 3.13 we have That 1 0 is a unit follows from requirement (m1).

Y. Ueda
The separate continuity of multiplication follows from that the E n and the ι m,n are all normal maps.
(2) We first confirm that (3.8) is well defined.We have which shows that the mapping given by (3.8) in question is well defined thanks to [17, Remark 3.1].It is obvious that the resulting map is bilinear.
by the above computation utilizing Corollary 3.13.Then, taking the limit as k → ∞ we have, by [17,Proposition 4.16], as long as > n.Taking the limit as → ∞ we conclude that This inequality shows that replacing Θ∞,n with Θ ∞,n in (3.8) still works to define a left Σ(α t , β)module structure on S(α t , β).
The last assertion is trivial from requirement (m1).
Here is an observation about the positivity on S(α t , β) and S(α t , β) in relation with the left Σ(α t , β)-module structure established above.
Proof .Let f ∈ Z(A m ) + and s = Θ∞,n (g) with g ∈ Z(A n ) be given.Assume s ≥ 0. Then Θ ,n (g) ≥ 0 in Z(A ) for some ≥ n thanks to the construction of S+ (α t , β) (see the place just before [17,Lemma 3.2]).Thus, Hence the former has been confirmed.We then confirm the latter by using the former.Let s ∈ S + (α t , β) be given.Then s = q S (s) with s = s * ∈ S(α t , β) holds.By the construction of S + (α t , β) (see the place just before [17, Lemma 3.10]) we observe that s + δë ≥ 0 for all δ > 0. Since The remaining question is what condition on the left multiplicative inductive sequence A n makes both S(α t , β) and S(α t , β) unital commutative algebras.We give such a condition below, though it is not satisfied in the case of U q (n), our main example.Actually, Σ(α t , β) is not commutative in the case.See Proposition 4.21.
Let us introduce the following property (c): For each pair m, n ≥ 0 there are γ The motivation for introducing the property (c) is to abstract the cases of infinite symmetric group S ∞ as well as infinite-dimensional unitary group U(∞) in the present framework.In those classical cases, each γ m+n is given by an inner conjugacy and each α t m+n must be the trivial flow.The next proposition shows that this property is sufficient to make both S(α t , β) and S(α t , β) unital commutative algebras.Proposition 3.16.If the left multiplicative structure ι m,n has property (c), then all Σ(α t , β), S(α t , β) and S(α t , β) are unital commutative algebras with units 1 0 , ë and e, respectively.Moreover, the mappings x ∈ Σ(α t , β) → x • ë ∈ S(α t , β) and x ∈ Σ(α t , β) → x • e ∈ S(α t , β) define surjective algebra homomorphisms, and furthermore, S(α t , β) is a normed algebra.
Proof .Let m, n ≥ 0 be fixed.For simplicity, we write Observe that for each z ∈ Z m+n , γ m+n induces an * -automorphism of zA m+n by (c1).Then, one can confirm, by checking the KMS condition thanks to (c1) and (c3), that τ z • γ m+n = τ z holds.

Y. Ueda
Let f ∈ Z(A m ) and g ∈ Z(A n ) be arbitrarily given.We have This immediately implies that Σ(α t , β) is a commutative algebra.It also follows that We have as long as m > m, and applying q S to this identity we also obtain that as long as m < m .In particular, defines a well-defined, commutative multiplication on S(α t , β).The proof of Theorem 3.14 (2) shows that as long as m > m.It follows, by taking the limit as m → ∞, that also defines a well-defined, commutative multiplication on S(α t , β).That ë and e become units of those algebra, respectively, are trivial.Also, the last assertion is clear now.Remark 3.17.Propositions 3.15 and 3.16 enable us to prove, by a standard method (see, e.g., [2, Proposition 4.4 and Exercise 4.2], that the so-called ring theorem holds for S(α t , β) if the left multiplicative structure ι m,n has property (c).

Olshanski's representation ring of U(∞)
Let A n = W * (U(n)) be the group W * -algebra of the unitary group U(n) of rank n, and consider the case when the flow α t is the trivial one because no q-deformation is necessary and dealing with tracial states is suitable in this case (see Section 4).We will investigate the algebra Σ := Σ(α t , β), the representation (operator) systems S = S(α t , β) and S = S(α t , β), all of which must be independent of the flow and the inverse temperature in the case.We also remark that every E n must be the center-valued trace in the present setup.
The consequence gives an operator algebraic point of view to Olshanski's rings R and R/(ϕ − 1) with the notation in [26].Actually, we will show that Σ and S are exactly the same as R and R/(ϕ − 1), respectively.In what follows, we will freely use the notation in [26, Sections 2 and 3.1-3.2],but n stands for the parameter of length of signature instead of N .
In the present case, we may and do identify Z n with the set S n of all signatures of length n.We will denote by z λ the corresponding minimal central projection for a λ ∈ S n .Then we have Z(A n ) ∼ = ∞ (S n ) by z λ → δ λ , where δ λ is the Dirac function taking 1 at λ and 0 elsewhere.By definition we also have R n ∼ = ∞ (S n ), as normed spaces, by sending each , where dim(λ) denotes the dimension of a representation with the signature λ, so that dim(λ) := dim(z λ ).Consequently, we have , where n≥0 denotes the algebraic direct sum over n ≥ 0.
We then investigate the multiplication on R. For each λ ∈ S m+n we have with rational Schur functions s λ .This indeed describes the expansion of the irreducible character of U(m + n) of label λ, whose restriction to the maximal torus is s λ , restricted to U(m) × U(n), via the embedding With this quantity c(λ | µ, ν), the multiplication on R is determined by See [26, equation (2.13)].
It is time to specify the left multiplicative structure on the sequence A n = W * (U(n)) that we are going to discuss.The embedding (3.10) gives a left multiplicative structure ι m,n : This fact implies that the structure ι m,n enjoys property (c) since the α t is assumed to be the trivial flow.Consequently, the resulting Σ, S and S are all unital commutative algebras by Proposition 3.16.We will look at the structure ι m,n in some detail below.

It follows that
Comparing (3.11) with (3.12) we observe that the vector space isomorphism R ∼ = Σ transplants the multiplication on R into that on Σ determined by which is identical to the multiplication on Σ introduced in Theorem 3.14 (1).Thus we have R ∼ = Σ as algebras.
Since ϕ k = σ (k) by [26, equation (2.9)] and dim((k)) ≡ 1, we have ϕ : commutes.Here Γ : S → C ex S ln (S) is defined by Γ(s)(ω) := ω(s) for any s ∈ S and ω ∈ ex S ln (S) .(This map is multiplicative by Remark 3.17.)This shows that the representation system S(α t , β) is a correct counterpart of representation ring.In this respect, we see that S ∼ = S, or other words, q S is injective, if we take [26, Proposition 3.9(iii)] into accounts with the help of U(∞) ∼ = ex S ln (S) thanks to Theorem 3.11.However, the mechanism behind this phenomenon is not so clear to us at the moment.
Remark 3.18 (the infinite symmetric group S ∞ = lim − → S n ).The framework we have developed so far can also be used for the group C * -algebra C * (S ∞ ), which is the special case when A n = C * (S n ) = CS n , finite-dimensional * -algebras.Consider the tracial states on C * (S ∞ ) so that α t must be trivial and all the E n become center-valued traces.By Remark 3.2, S ∼ = K 0 (C * (S ∞ )) ⊗ Z C, and hence the study of S (or S) is exactly the same as that of the dimension group of C * (S ∞ ) due to Vershik-Kerov.
Consequently, the present framework generalizes both Vershik-Kerov's use of dimension groups as well as Olshanski's representation ring of U(∞) in a unified way.Moreover, we will explain, at the end of this paper, how the framework is applied to the quantum group case.

Quantum group setup
In this section, we apply the theory we have developed so far to inductive sequences of compact quantum groups.In fact, the notion of (α t , β)-spherical representations originally comes from a consideration of quantum groups.A short comment on this was already given at [35,Remark 9.4].

Compact quantum groups
We first consider single compact quantum groups to fix the notation and to collect necessary facts on them.To the best of our knowledge, no analytical attempts have been made to spherical unitary representations (rather than spherical functions) for (Gelfand) pairs G < G × G (via the diagonal embedding) even in the compact quantum group case.The discussions on compact quantum groups below entirely follow Neshveyev-Tuset's book [18, Chapters 1 and 2] (a standard reference on the operator algebra side), but we sometimes use slightly different symbols, such as "id" (instead of "ι" there) denoting the identity map, from their book in order to keep the notation so far.
Let G be a compact quantum group, which means a Hopf * -algebra (C[G], ∆, S, ε) that satisfies the assumption of [18,Theorem 1.6.7].Then, we have the "dual Hopf * -algebra" U(G), ∆, Ŝ, ε , whose precise definition is given in the final paragraph of [18,Section 1.6].Especially, one has to notice that the comultiplication ∆ is a The general principle of Hopf algebra duality suggests that U(G) plays a rôle of group algebra of G, and the "group However, this pair of * -algebras is too big to discuss its unitary representation theory, and hence we develop the theory in terms of W * (G).We will clarify what W * (G) is by supplying a few facts, because [18] does not touch it explicitly.The explanation below explicitly or implicitly uses Yamagami's idea [40].
Any unitary representation U ∈ B(H U )⊗C[G] on a finite-dimensional Hilbert space H U gives a * -representation π U : U(G) H U by π U (x) := (id ⊗ x)(U ) for any x ∈ U(G).Choose and fix representatives U z , z ∈ Z, of the equivalence classes of irreducible unitary representations of G throughout.The U z , z ∈ Z, enables us to construct a bijective * -homomorphism One of the U z must be the trivial representation, and we denote by 1 the corresponding z.
Here is the definition of group W * -algebra W * (G) (and its natural σ-weakly dense * -subalgebra F(G)).Our spherical unitary representation theory will be constructed based on W * (G) rather than C[G].

Definition 4.1 ([40]
).The group W * -algebra W * (G) is defined to be all the x ∈ U(G) such that sup z∈Z π Uz (x) < +∞.Let F(G) be the * -subalgebra consisting of all x ∈ U(G) such that π Uz (x) = 0 for all but finitely many z.The group C * -algebra C * (G) is defined to be the norm-closure of F(G).
Both F(G) ⊂ W * (G) do not depend on the choice of representatives U z , though the above definition itself does.Moreover, one easily sees that x ∈ W * (G) if and only if there exists a C > 0 so that π U (x) ≤ C for any finite-dimensional unitary representation U .Observe that in the W * -algebra or ∞ -sense by the bijective * -homomorphism (4.1), and hence W * (G) is an atomic finite W * -algebra.
Similarly, F(G) is isomorphic to the algebraic direct sum z∈Z B(H Uz ) by (4.1) and becomes a σ-weakly dense (non-unital) * -subalgebra of W * (G).In what follows, we denote by z the central projection supporting the direct summand corresponding to each z ∈ Z so that z U ) with the leg notation (n.b., U 13 V 24 is the "outer tensor product" of U and V ).As before, we have a bijective * -homomorphism The tensor product W * -algebra and identified with all the x ∈ U(G×G) with sup (z 1 ,z 2 )∈Z×Z (π Uz 1 ⊗π Uz 2 (x) < +∞.We denote by where the T ∈ Mor(W, U × V ) are selected by the (unique) irreducible decomposition U × V ∼ = W W . (See [40] too in this respect.)Hence, the restriction of ∆ to W * (G) defines a normal comultiplication ∆ : W * (G) → W * (G × G).We also remark that It immediately follows that the restriction of ε to W * (G) gives a normal * -character.
There is a special positive element ρ ∈ U(G) so that and R(x) := ρ −1/2 Ŝ(x)ρ 1/2 for all x ∈ U(G) defines the unitary antipode on U(G).See [18, Section 1.7] for these facts and the interpretation of functional calculus We have an action ϑ ζ := Adρ iζ : C U(G).Since ρ it falls in W * (G), it induces a flow ϑ t : R W * (G), which is clearly continuous in the u-topology (see, e.g., [35,Lemma 7.1] for the topology) and fixes elements of the center Z(W * (G)).Since the second formula in (4.5) leads to we have Using [18,Example 2.2.22] we see that R defines an involutive * -anti-automorphism on W * (G) and that Ŝ(F(G)) = F(G) (but Ŝ(W * (G)) ⊆ W * (G) does not hold in general).Moreover, both Ŝ and R send each direct summand z U(G) = zW * (G) ∼ = B(H Uz ) to the one z U(G) = zW * (G) ∼ = B(H Uz ) by (4.1), where z ∈ Z is a unique element such that U z is (unitarily) equivalent to the conjugate representation Ūz of U z (see [18,Definition 1.4.5]).In this way, we have an involutive bijection z → z on Z so that Ŝ(z) = R(z) = z (4.9) holds for any z ∈ Z as a minimal central projection of W * (G).Consequently, we have obtained a sextuplet W * (G), ∆, R, ρ it , ϑ t , ε that enjoys all the properties of [40, Definition 1.2] with encoding M := W * (G), ∆ := ∆, τ := R, u t := ρ it/2 , θ t := ϑ t/2 , ε := ε and M := F(G) in the notation there.As remarked there, this is a specialized (and even strengthened) version of Masuda-Nakagami's formalism [16] so that there are no (essential) differences between Sato's understanding of compact quantum groups in [31,32,33] and ours.Thus, his works are available here and so is the present work in his framework.
Here is a technical lemma.
H Π be a bi-normal * -representation.Then there are a unique bi-normal * -representation Π : H Π and a unique normal commutes, where the vertical arrows mean the natural embeddings thanks to W * (G × G) = W * (G) ⊗ W * (G) for the lower one.
Proof .The existence of desired Π is trivial by the construction of maximal C * -tensor products.
Observe that the family z 1 ⊗ z 2 with z 1 , z 2 ∈ Z is a complete family of minimal projections of Z(W * (G × G)) thanks to (4.2).Since zW * (G) ∼ = B(H z ) with dim(H z ) < +∞ for every z ∈ Z, we have for every z 1 , z 2 ∈ Z. Hence we have a normal * -representation By construction, Π clearly agrees with Π on W * (G) ⊗ W * (G) thanks to the bi-normality of Π. Hence we are done.
In what follows, we call a bi-normal * -representation Π : and Π ∆(x) ξ = ε(x)ξ holds for every x ∈ W * (G).We call such a vector a spherical vector in H Π of G < G × G through Π.
The next proposition, which is motivated by [14,Proposition 3.4], shows that the notion of spherical unitary representations of G < G × G is exactly equivalent to that of (ϑ t , −1)-spherical representations of W * (G).Proposition 4.4.For a unitary representation Π of G × G on H Π and a vector ξ ∈ H Π , the following are equivalent: See [35, Definition 5.1, Remark and Definition 5.2] for the notion of (ϑ t , −1)-spherical vectors.
Proof .(i) ⇒ (ii): Let x ∈ F(G) be arbitrarily given.Then we have with the multiplication map m.Thus, we have Observe that holds for any z 1 , z 2 ∈ Z thanks to (4.9).Since (z where the final equality is justified thanks to Ŝ(x) ∈ F(G) ⊂ W * (G) due to the assumption that by using (4.9).Hence we obtain that Substituting the unit 1 for x in the above identity implies Π(z 1 ⊗ z 2 )ξ = 0 if z 1 = z2 .Thus, we obtain that ξ = z∈Z Π(z ⊗ z)ξ, and consequently, (ii) ⇔ (iii): This was essentially explained in [35,Remark 9.4].In fact, one has and hence for all x ∈ F(G).Since F(G) is a σ-weakly dense * -subalgebra of W * (G), a standard approximation technique (see the proof of [35,Proposition 4.3]) enables us to confirm the desired equivalence based on the above two identities.
The last assertion was proved in the above proof of (ii) ⇒ (i).Hence we are done.

Y. Ueda
We give a complete classification of irreducible spherical unitary representations of G < G×G.The consequence is no surprise and the essentially same as the classical case.

Theorem 4.5. Any irreducible spherical unitary representation of
H Ū ⊗ H U for some irreducible unitary representation U of G, and any spherical vector in π Here, (R U , RU ) is a unique solution of the conjugate equations for U and Ū .
Proof .By (4.2), any irreducible normal representation of By the final assertion of Proposition 4.4 z 1 must be z2 , and hence π Uz 2 ⊗ π Uz 2 .Consequently, any irreducible spherical unitary representation of Using [35,Theorem 6.1] through Proposition 4.3(iii) we see that any spherical vector ξ is unique up to scalar multiple.Hence, thanks to Proposition 4.4(ii) and [18,Example 2.2.23] with the notation T ∨ there, it suffices to confirm that (T ∨ ⊗1)R U = (1⊗T )R U holds for any T ∈ B(H U ).However, this identity is known to be the characterization of map T → T ∨ ; see the proof of [18, Proposition 2.2.10].Hence we are done.

The above theorem says that any irreducible spherical unitary representation of G < G×G
can be captured by means of U(G × G) and its "spherical state" must be of the form x) holds true for any x, y ∈ F(G); see [18,Example 2.2.3].This should be regarded as the origin of quantized characters in the sense of Sato.

We have an anti-automorphism Θ on
and hence Θ ∆(U(G) = ∆(U(G)) holds.This suggests that ∆(U(G)) ⊂ U(G × G) is a realization of "quantum Gelfand pair" associated with G < G × G.However, Θ is not well defined on W * (G × G).
3. Item (1) above suggests us to "extend" the notion of spherical unitary representations to obtain the whole information of conjugate solutions.For the purpose, we remark that general quantum groups are not co-commutative so that ∆cop := σ • ∆ with the flip map σ gives another possible analog of the diagonal embedding g → (g, g) of ordinary groups.
Here is a trivial translation of Proposition 4.4: In the same setup there, the following are equivalent: Then any irreducible spherical unitary representation of G < G × G with replacing ∆ with ∆cop and any spherical vector in it must be of the form π U ⊗ π Ū : W * (G × G) H U ⊗ H Ū and a scalar multiple of RU (1) ∈ H U ⊗ H Ū , respectively.(See [18, Theorem 2.2.21] and its preliminary discussion.)Consequently, the R-part of solutions of conjugate equations can be obtained as (ϑ t , +1)-spherical representations.It may be possible to give further deformations of diagonal embedding ∆ : U(G) → U(G × G).

4.
There is no special reason, in Sato's approach [31,32] to asymptotic representation theory for quantum groups, for which choice β = ±1 of inverse temperature of KMS states is appropriate.(Note that only sign, or the time direction, is important, because the general β case can be transformed into one of the cases of β = ±1 by positive scaling.)Hence Sato's choice of β was just on an arbitrary basis and not essential.However, the spherical representation theory shows that the choice must be β = −1 and taking time reversal corresponds to the co-opposite transition (or the interchange of R U and RU in conjugate solutions).

Inductive limits of compact quantum groups
Let G n , n ≥ 0, be an inductive sequence of compact quantum groups with G 0 the trivial one, that is, for each n there is a surjective Hopf * -algebra morphism We have to formulate their inductive limit "G = lim − → G n " in the category of "quantum groups" to discuss spherical unitary representations of G < G × G.
The most widely accepted approach to "quantum groups" is based upon the notion of quantized universal enveloping algebras.The approach looks purely algebraic, but in principle, it involves differential operators, so that it requires a "differential structure" on infinite-dimensional (virtual) space G = lim − → G n .Such a structure has not completely been established so far even in the classical setting such as U(∞) = lim − → U(n), to the best of our knowledge.Thus, it is difficult to use quantized universal enveloping algebras directly for our purpose.See Section 4.4.6 for a related discussion.Moreover, the most plausible approach to unitary representations is the use of group C * -algebras.However, the lack of "Haar measures" even in the classical setting due to the infinite-dimensionality becomes a serious issue to use the standard construction of group C * -algebras.
With these reasons, Sato [32] used the framework of W * -bialgebras to understand G and developed its asymptotic unitary representation theory of Vershik-Kerov's type based on his earlier work [31].The approach to formulate G below is slightly different from his (and hopefully more natural than his), though they are equivalent at least for the questions he considered.

Inductive Baire group C * -algebra
) that respects the dual Hopf * -algebra structure.We can take the algebraic inductive limit U(G) := lim − → U(G n ) (see, e.g., [9,Chapter 2] for the notion of inductive (or direct) limits), which is too big to discuss unitary representations.Here, we observe that the embedding sends W * (G n ) into W * (G n+1 ).Thus we have an inductive sequence W * (G n ) with the following additional structures: comultiplications ∆n : Those structures are well behaved under the embeddings W * (G n ) → W * (G n+1 ) that come from the natural ones U(G n ) → U(G n+1 ); namely, the restrictions of ∆n+1 , εn+1 , Rn+1 and ϑ t n+1 to W * (G n ) are exactly ∆n , εn , Rn and ϑ t n , respectively.Here is a remark.Since the special positive elements ρ n (see (4.4)) do not form an inductive sequence in any sense, it is a bit non-trivial that the compatibility of flows ϑ t n as well as the Y. Ueda unitary antipodes Rn with the embeddings U(G n ) → U(G n+1 ).This fact can be confirmed by using the compatibility of antipodes Ŝn as follows.For any x ∈ U(G n ), we have by the definition of unitary antipodes Rn with the help of (4.6).It follows that [ρ n , ρ n+1 ] = 0 = ρ −1 n ρ n+1 , x for all x ∈ U(G n ).Hence we have for all x ∈ U(G n ).This and (4.7), (4.8) imply the compatibility of unitary antipodes Rn .We remark that this argument clearly works for any pair of compact quantum group and its subgroup.So far and in what follows, we do not use any symbols to specify the embedding of W * (G n ) into W * (G n+1 ) unlike [31] for simplicity, though it encodes an important representation theoretic information; in fact, the embedding comes from the restriction procedure for irreducible unitary representations of G n+1 to G n .See Lemma 4.14.
In the above setup, we consider the C * -algebraic inductive limit B(G) := lim − → W * (G n ) and denote by B ∞ (G) its canonical local W * -subalgebra (that is the algebraic inductive limit of W * (G n )'s).We think of this C * -algebra as a group algebra corresponding to a kind of "inductive Baire structure" associated with G = lim − → G n .We also consider the C * -algebraic inductive limit Clearly, all the maps above are locally normal.The pentad B(G), ∆ : B(G) → B(G×G), R, ϑ t , ε enjoys almost all the requirements to be "Hopf * -algebra" like (U(G n ), ∆n ).
Let W * (G) be the locally normal enveloping W * -algebra of B(G), i.e., a unique W * -algebra whose predual is given by all the locally normal linear functionals on B(G) so that B(G) sits in W * (G) as a σ-weakly dense subalgebra.Similarly, let W * (G × G) be the locally normal enveloping W * (G) ⊗ W * (G) (whose existence is an easy exercise), we have obtained a normal * -homomorphism which is nothing but the comultiplication in [31].In this way, Sato formulated G = lim − → G n within the framework of W * -bialgebras.Although his approach has a merit as an attempt to enlarge the operator algebraic framework of quantum groups beyond the locally compact setting, we prefer to regard ∆ : B(G) → B(G × G) as a "true comultiplication" instead.

Spherical unitary representations
It is easy to see that the algebraic tensor product B ∞ (G) ⊗ B ∞ (G) is naturally identified with the algebraic inductive limit of the sequence We will translate Lemma 4.2, Definition 4.3 and Proposition 4.4 above into Lemma 4.8, Definition 4.9 and Proposition 4.10, respectively, for the inductive limit H Π be a locally bi-normal * -representation.Then, there are a locally bi-normal * -representation Π : H Π and a locally normal commutes, where the vertical arrows mean the embeddings explained above.
Proof .It is obvious that Π extends to B(G) ⊗ B(G), and hence does to B(G) ⊗ max B(G) by means of maximal C * -algebraic tensor products.This extension is nothing but the desired Π.
For each n, we consider the restriction Π n of Π to W * (G n ) ⊗ W * (G n ) and apply Lemma 4.2 to this restriction.Then we have a unique normal * -representation Π n : W * (G n × G n ) H Π that agrees with Π n .Thanks to the normality, we see that Π n+1 agrees with Π n over the whole We call a locally bi-normal * -representation Π : The above lemma shows that there are two natural extensions Π : Here is a definition.
It is now easy to confirm that this definition completely agrees with the definition of (ϑ t , −1)spherical representation.Namely, all the results in our previous paper can successfully be applied to spherical representations of G < G×G with G = lim − → G n .In other words, our studies certainly justify Sato's asymptotic representation theory for G = lim − → G n from the viewpoint of spherical representations, and especially, explains why his notion of quantized characters playing a central rôle in [31,32,33] is appropriate.

Y. Ueda
Proposition 4.10.Let Π be a unitary representation of G × G on a Hilbert space H Π and ξ ∈ H Π be a non-zero vector.The following are equivalent: See [35, Definition 5.1, Remark and Definition 5.2] for the notion of (ϑ t , −1)-spherical vectors.
Proof .(i) ⇔ (ii): Since ∆ = lim − → ∆n and ε = lim − → εn together with density, item (i) is equivalent to that Π ∆n (x) ξ = εn (x)ξ holds for all x ∈ W * (G n ×G n ) and n ≥ 0. Applying Proposition 4.4 to each level n, we see that this is equivalent to item (ii).
(ii) ⇔ (iii): As in the proof of Proposition 4.4(ii) ⇔ (iii) we see that item (ii) is equivalent to Using the σ-weak density of together with the Phragmen-Lindelöf method, we can prove that ξ is a (ϑ t , −1)-spherical vector in Π Let Π be a unitary representation of G × G with G = lim − → G n on a Hilbert space H Π .By construction we observe that on H Π , and hence [35, Theorem 6.1] is available in this setup thanks to the above proposition.Therefore, we conclude that G < G × G is a Gelfand pair in the sense of Olshanski; see, e.g., [35,Section 6].
The above proposition also gives the next desired assertion.
Corollary 4.11.Let Π be a unitary representation of G × G with G = lim − → G n on a Hilbert space H Π and ξ ∈ H Π be a unit vector.Then the following are equivalent: Remark that if the ∆n are replaced with ∆cop n as in Remarks 4.6(3), then one obtains locally normal (ϑ t , +1)-spherical representations.
The above corollary guarantees that the general theory for (α t , β)-spherical representations developed in the previous and this papers can be applied to spherical unitary representations of op .Hence we first have to show that no differences occur among those choices of algebras to discuss unitary representations of G × G.The next lemma guarantees that one can use each of those algebras to examine the unitary equivalence for unitary representations of G × G. Lemma 4.12.Let Π i be a unitary representation of G × G on a Hilbert space H Π i for each i = 1, 2. For any u ∈ B(H Π 1 , H Π 2 ) the following are equivalent: Then, by the norm-continuity we obtain item (ii).
Π χ , H χ , ξ χ µ(dχ), and hence holds for all x, y ∈ B ∞ (G).Since the σ-weak density of each together with the local normality of Π χ plus the Kaplansky density theorem and then since the norm-density of n≥0 W * (G n × G n ) in B(G × G), we can easily confirm that χ → Π χ (x) is Borel for every x ∈ B(G × G), and obtain that In this way, all the formulations Π, Π, Π and Π rop of a spherical unitary representation of G < G × G admit a simultaneous spectral decomposition with a unique probability measure over the extreme points ex K ln −1 (ϑ t ) .

Quantized character functions
So far, we have provided all the necessary ingredients to discuss the (spherical) unitary representation theory for G = lim − → G n .As a consequence, we saw that the study of (spherical) unitary representations comes down to that of locally normal (ϑ t , −1)-KMS states.Thus, one may regard such a state as a quantum analog of character, and this idea was indeed employed by Sato in [31] (and his later works) to develop the asymptotic representation theory for quantum groups (without appealing to spherical representations).In our opinion, it is more appropriate to formulate quantized characters as "functions over G".This aspect was already considered by Sato [31,Section 3.3] based on Arano's suggestion to him.However, Sato's discussion there is not entirely satisfactory (at least to us), because his discussion depends on Gorin's work [11] and is performed only in a special case.Thus, we give here a new formulation of quantized characters of G, which enables us to give a (hopefully nice) interpretation to Gorin's work in terms of quantum groups in the next section.
Let C(G n ) be the C * -enveloping algebra of C[G n ].See [18,Section 1.7], where the symbol x n for all n ≥ 0 , in which we can construct a "function" associated with any χ ∈ K ln −1 (ϑ t ).This type of algebras was considered in [15] to understand the quantized universal gauge group SU q (∞) in Woronowicz's formalism.We remark that if all the G n are ordinary groups, then C(G) is exactly identified with the algebra of continuous functions over the topological group G = lim − → G n .
We will use the bijective * -isomorphism Φ U• ; see (4.1).We define a unitary element naturally.It is easy (see the proof of the lemma below) to confirm that this unitary element u z ∈ zW * (G n ) ⊗ C[G n ] depends only on the unitary equivalence class of U z .Here is a technical lemma.Lemma 4.14.For every z ∈ Z n+1 , we have The above identity is essentially the irreducible decomposition of the restriction of U z to G n .The reason why the information of multiplicities does not appear in the above identity is that it is built in the embedding W * (G n ) → W * (G n+1 ).Actually, the multiplicity of U z in U z is exactly Tr(zz ) with a (unique) non-normalized trace Tr on zW * (G n+1 ) ∼ = B(H Uz ); this is consistent with Bratteli diagram description of the inductive sequence W * (G n ), see Section 2 or [35,Section 9] in detail.We remark that the essentially same observation has been given in [34,Lemma 2.10] in a slightly different language.
Proof .It is convenient here to write the embedding map of and it also induces the embedding Note that zU(G n+1 ) = zW * (G n+1 ) by definition.We will compute the both sides of the above desired identity in the following framework: Here, L(V, W ) stands for all the linear maps from a vector space V to another W .The horizontal embeddings are given by (a⊗b)x = x(b) a for any element x of U(G n+1 ) and U(G n ), respectively.We first remark that In fact, for every x ∈ U(G n+1 ), we have , where we regard B(H Uz ) naturally as before.
Since zW * (G n+1 ) is finite dimensional, we see that z t r n+1 n (z ) = 0 for only finitely many z ∈ Z n .Then, for every y ∈ U(G n ) we have . This shows the desired identity.
Let ω ∈ S ln (B(G)), the convex set of all locally normal states on B(G), be arbitrarily given.
and since z∈Zn ω(z) = 1 thanks to the local normality of ω, we see that converges in C(G n ) with respect to the norm topology.This should be understood as a generalization of [11,Proposition 4.6].Namely, the discussion here enables us to give a representation-theoretic interpretation to Gorin's q-Schur generating functions; in fact, such a function is nothing less than the restriction of a quantized character function to the "diagonal subgroup" of U q (∞); see the next section.
Proof .We have as before, we have thanks to the local normality of ω.Hence we are done.This formulation of quantized characters is different from [31] and [32].The next proposition enables one to compute any quantized character functions explicitly by utilizing the representation theory of G n .Proposition 4.17.Every quantized character function χ ) admits the following expression: where ρ n denotes the special positive element of U(G n ) (see (4.4)) and Tr stands for the nonnormalized trace on B(H Uz ).
Proof .We have χ . These three facts immediately imply the desired assertion.Actually, we have Hence we are done.
The above proposition says that the right-hand side of (4.11) depends only on the unitary equivalence class of unitary representation U z , and moreover, that any quantized character function χ(u) can explicitly be computed in principle by the data χ(z), z ∈ Z := n≥0 Z n , and the representation theory of each G n that is enough to compute (4.11) explicitly.Proof .We first claim that h((τ  Assume that χ i → χ in K ln −1 (ϑ t ) with respect to the weak * -topology (n.b., the weak * -topology is metrizable; see [35,Corollary 8.3]).Let n ≥ 0 be arbitrarily fixed and choose an increasing sequence F k of finite subsets of Z n .We have [35,Proposition 7.10].Thus, we have shown that χ → χ(u) is open.
We can discuss the multiplication as well as the adjoint operations for quantized character functions in C(G), which will be translated in terms of K ln −1 (ϑ t ) in the next proposition that contains [32, Theorem 4.1] essentially.1.For any pair ω 1 , ω 2 ∈ S ln (B(G)), there is a unique locally normal state In particular, K ln −1 (ϑ t ) becomes a semigroup with multiplication (χ 1 , χ 2 ) → χ 1 • χ 2 and neutral element ε, and the mapping χ → χ(u) is an injective, semigroup homomorphism from ) and χ * (u) = χ(u) * holds.In particular, the family of quantized character functions is closed under the adjoint operation if and only if ϑ t is a trivial flow.
(2) Since ∆ = lim − → ∆n is clearly locally normal and (4.7), one can easily confirm that χ 1 • χ 2 falls into K ln −1 (ϑ t ).Let n ≥ 0 and z ∈ Z n be arbitrarily chosen and fixed.By (4.3) we have The discussion in this section clearly works for single compact quantum groups.It is also (probably) possible to give a similar formulation of spherical functions of G < G × G.This will be discussed elsewhere in a wider setup.

A concrete example: U q (∞)
We will examine a concrete example, which was already investigated by Sato [31,32], from our viewpoint.In what follows, we choose and fix 0 < q < 1 throughout.4.4.1 Quantum unitary group U q (n) We begin by making it clear what the quantum unitary group U q (n) we employ is.
Let U q gl(n) be the Drinfeld-Jimbo quantized universal enveloping algebra associated with the general linear Lie algebra gl(n).Namely, it is the unital algebra generated by the Cartan-type elements K ±1 i , 1 ≤ i ≤ n and the elements x j , y j , 1 ≤ j ≤ n − 1, corresponding to the standard positive and negative simple root vectors of gl(n) with the relations given in [21, Section 1.2].The algebra U q gl(n) becomes a Hopf * -algebra with the following structure: See, e.g., [21,Section 1.2].Applying [18,Theorem 2.3.13] to this Hopf * -algebra with its finitedimensional type 1 * -representations, we obtain the compact quantum group U q (n) with Hopf * -algebra (C[U q (n)], ∆ n , S n , ε n ), which is exactly the coordinate ring A(U q (n)) of the so-called quantum unitary group of rank n discussed, e.g., in [19, Section 1], [20] (see [7,Section 2] for its concise review) in the context of q-analysis.(Note that the choice of generators of U q gl(n) here follows [19] (rather than [20]) with K ±1 i := q ± i , x j := e j , y j := f j .)As in Section 4.1 we then obtain the Hopf * -algebra U(U q (n)), ∆n , Ŝn , εn and the group W * -algebra W * (U q (n)) ⊂ U(U q (n)).Note that there is a canonical embedding U q gl(n) → (U q gl(n)) * * → C[U q (n)] * = U(U q (n)) as Hopf * -algebras.Thus, we may and do regard U q gl(n) as a Hopf * -subalgebra of U(U q (n)), and hence obtain as Hopf * -algebras, that is, ∆n,q , Ŝn,q , εn,q are just the restrictions of ∆n , Ŝn , εn , respectively.Hence we will use only the symbols ∆n , Ŝn , εn via the above embedding.An important thing is that the representation tensor categories (with conjugates) of U q gl(n) and U q (n) are naturally identified with each other; see [18,Definition 2.4.5].Hence the algebras W * (U q (n)) ⊂ U(U q (n)) can directly be constructed from U q gl(n) (or more precisely, its representation category).Another important fact is that the special positive element ρ n ∈ U(U q (n)) (see (4.4)) is given by ).These facts show that all essential ingredients used in the present theory can directly be calculated in terms of U q gl(n), though our theory is developed based on group W * -algebras W * (U q (n)).
The minimal projections Z n of W * (U q (n)) are known to be labeled by the signatures S n of length n, and thus we will use the signatures S n as an index set instead of Z n itself.Hence the natural mapping λ ∈ S n → z λ ∈ Z n will be used in what follows.

Infinite-dimensional quantum unitary group U q (∞)
Let us consider the standard embedding U q gl(n) → U q gl(n + 1) by sending the first n Cartantype elements K ±1 i 's and the first n − 1 positive and negative simple root vectors x j , y j 's to the corresponding ones (with the same indices).By the construction of C[U q (n)], the restriction map from (U q gl(n + 1)) * to (U q gl(n)) * via U q gl(n) → U q gl(n + 1) induces a surjective Hopf * -algebra morphism C[U q (n + 1)] C[U q (n)].Hence we can take the inductive limit U q (∞) = lim − → U q (n) in the sense of Section 4.2.Namely, we have where the first algebra is a C * -algebraic inductive limit, while the last three algebras are algebraic inductive limits.We also have B(U q (∞) × U q (∞)) := lim − → W * (U q (n) × U q (n)) and ∆ = lim − → ∆n : B(U q (∞)) → B(U q (∞) × U q (∞)), R = lim − → Rn , ϑ t = lim − → ϑ t n , ε = lim − → εn as in Section 4.2.

Diagonal subgroup of U q (∞)
Let U q h(n) be the unital * -subalgebra generated by the K ±1 i , 1 ≤ i ≤ n.Clearly, it is a commutative, co-commutative Hopf * -subalgebra of U q gl(n).Let C[T n ] be the unital * -algebra consisting of all the restrictions of elements of C[U q (n)] (⊂ (U q gl(n)) * ) to U q h(n) so that we have a surjective Hopf * -algebra homomorphism | Tn : C[U q (n)] C[T n ] that extends to the C * -level | Tn : C(U q (n)) C(T n ) = C(T n ) sending the canonical generators t ij and det −1 q to δ i,j t i and (t 1 • • • t n ) −1 , respectively, or determined by the paring = q i m i m i , where C(T n ) denotes the algebra of continuous functions on the n-torus T n and t i ∈ C(T n ) does the ith coordinate function on T n .
It is well known that the following diagram commutes: C(U q (n + 1)) C(U q (n)) Thus there is a unital * -homomorphism | T∞ from the σ-C * -algebra C(U q (∞)) = lim ← − C(U q (n)) to the σ-C * -algebra C(T ∞ ) = lim ← − C(T n ) by f = (f n ) → f | T∞ := (f n | Tn ).We call this unital * -homomorphism | T∞ : C(U q (∞)) → C(T ∞ ) the diagonal subgroup of U q (∞).4.4.4Gorin's q-Schur generating functions and quantized character functions Theorem 4.13 shows that all the unitary equivalence classes of spherical unitary representations of U q (∞) and all the locally normal (ϑ t , −1)-KMS states K ln −1 (ϑ t ) are in one-to-one correspondence.Thus, we can label such a unitary equivalence class with a unique element of K ln −1 (ϑ t ).This certainly justifies Sato's formulation of asymptotic representation theory of U q (∞) as mentioned before.

Smooth quantized characters
The short discussion below is just a remark to give a basis for future research.
unitary representation of G × G. Then we have a unique bi-normal * -representations Π : W * (G) ⊗ max W * (G) H Π and a unique normal * -representations Π : W * (G × G) H Π as in the above lemma.Y. Ueda Definition 4.3.A unitary representation Π of G × G on a Hilbert space H Π with a unit vector

Definition 4 . 7 .
We call B(G) the inductive Baire group C * -algebra of G = lim − → G n and understand the pentad B(G), ∆ : the sense of Takeda.(See [35, Section 5.2] for the notion of locally normal enveloping W * -algebras and W * -inductive limits.)We have normal extensions of ε, R, ϑ t to W * (G) and also a normal * -homomorphism ∆ : W * (G) → W * (G × G) with keeping the same symbols.With a canonical surjection W * (G × G) Thus the inductive structure of B(G × G) enables us to construct a locally normal * -representation Π : B(G × G) H Π .By construction, this * -representation agrees with Π on every W * (G n ) ⊗ W * (G n ) so that the commutative diagram holds.Hence we have obtained the desired locally normal * -representation Π.
We call this ω(u) ∈ C(G) the quantized function on G associated with ω.Namely, each ω ∈ S ln (B(G)) defines a quantized function ω(u) ∈ C(G), and its projection ω(u n ) ∈ C(G n ) should be understood as the restriction of ω(u) to G n .Definition 4.16.The quantized function χ(u) ∈ C(G) associated with any χ ∈ K ln −1 (ϑ t ) is called a quantized character function of G.