Multiplicative characters and Gaussian fluctuation limits

It is known that extreme characters of several inductive limits of compact groups exhibit multiplicativity in a certain sense. In the paper, we formulate such multiplicativity for inductive limit quantum groups and provide explicit examples of multiplicative characters in the case of quantum unitary groups. Furthermore, we show a Gaussian fluctuation limit theorem using this concept of multiplicativity.


Introduction
Asymptotic representation theory initiated by Vershik and Kerov is a unitary representation theory of inductive limits of compact groups and a significant framework to study characters of such groups (see [37,38,39] etc.).For instance, using Vershik-Kerov's idea, we can describe complete lists of the extreme characters of various inductive limit groups.In particular, the characters of the infinite symmetric group S(∞) and the infinite-dimensional unitary group U(∞) are well studied and have intimate connections with many branches of mathematics (see [5,6,7,8,9,18,19,28] etc.).One of the features of extreme characters of S(∞), U(∞) is that they are multiplicative (see Section 3.4).Voiculescu [43] first noticed this fact for U(∞) based on the theory of type II 1 factors.After that, Vershik and Kerov showed that the extreme characters of S(∞) are also multiplicative.In Vershik-Kerov's work [20,40,41,42], the dimension group (i.e., K 0 -group) of the group C * -algebra C * (S(∞)) plays an important role.More precisely, there exists a bijection between the characters of S(∞) and the states of the dimension group, and the states corresponding to extreme characters are multiplicative, where the dimension group admits a natural ring structure.Moreover, this fact implies that the extreme characters of S(∞) are multiplicative.However, inductive limits of compact groups are not generally locally compact, and hence the meaning of their group C * -algebra are unclear.Therefore, one can not directly extend Vershik-Kerov's idea to general settings.Nevertheless, Boyer [10] introduced an appropriate ring for an inductive system of compact groups.For U(∞), Olshanski [27] studied such a ring, called the representation ring, based on the theory of symmetric functions.The representation ring plays the same role as the dimension group for S(∞), and one can show that the extreme characters of U(∞) are multiplicative by Vershik-Kerov's approach.
The author [31,32] initiated the asymptotic representation theory for quantum groups.Our formulation of the theory, among other things, fits Gorin's study of the q-deformed Gelfand-Tsetlin graph in [17].Thus, based on Gorin's work, one can obtain a complete list of the extreme quantized characters of the infinite-dimensional quantum unitary group U q (∞), where U q (∞) is the inductive limit of the quantum unitary groups U q (N ).
In the paper, we will proceed with investigating quantized characters of U q (∞) and particularly study their multiplicativity (in the sense of Definition 2.3).Recently, Ueda [35,36] extended the notion of the representation ring to more general settings.Thus, based on his idea, one can study the multiplicativity of quantized characters similarly to the classical case.In particular, we give concrete examples of multiplicative quantized characters of U q (∞).
We will also give an application of multiplicative characters to probability theory.As mentioned above, the character theory of U(∞) relates to various branches of mathematics.For instance, Borodin and Bufetov discovered that specific characters of U(∞), called one-sided Plancherel characters, produce Gaussian fields (see [3,4]).Their work gives a type of central limit theorem on the universal enveloping algebras of U(N ).In the paper, we show a central limit theorem on the group von Neumann algebras of U(N ) (see Corollary 4.3).Our result is a new type of Gaussian limit theorems in asymptotic representation theory, but our idea is essentially the same as in studies of a central limit theorem for quantum spin chains (see [12,13,14,15,16,23]). Similar to those work, fluctuations of non-commutative random variables are described by quasi-local algebras, and we can obtain an explicit algebraic description for the Gaussian limits of those fluctuations using Weyl CCR algebras and quasi-free states.
The organization of the paper is the following: In Section 2, we discuss KMS states on a quasilocal algebras with a flow.The main theorem here gives a necessary and sufficient condition that KMS states become multiplicative in the sense of Definition 2.3 (see Theorem 2.4).If a quasi-local algebra additionally has shift translations (see Section 2.2), then we introduce a representation ring of the quasi-local algebras.In Theorem 2.9, multiplicative KMS states are characterized in terms of this representation ring.In Section 3, we apply the general theory in Section 2 to the quasi-local algebra given by quantum unitary groups.As a result, we give concrete examples of multiplicative quantized characters of U q (∞) (see Proposition 3.7).In Section 4, we give a Gaussian fluctuation limit theorem with respect to multiplicative characters of U(∞) (see Theorem 4.1, Corollary 4.3).

Multiplicative states on quasi-local algebras 2.1 Quasi-local algebras
In this section, we discuss KMS states on a quasi-local algebra with flow.In Theorem 2.4, we give a necessary and sufficient condition that KMS states become multiplicative in the sense of Definition 2.3.Throughout this section, we denote by I the set of all finite intervals of N or Z.
Let M be a unital C * -algebra and (M I ) I∈I a family of W * -subalgebras in M, that is, M I is a C * -subalgebra of M and there exists a unique Banach space (M I ) * such that (M I ) * Let (M, (M I ) I∈I ) be a quasi-local algebra.We now follow the setup in [35,36].Namely, we assume that M has a flow α : R ↷ M such that α t (M I ) = M I and lim t→0 ∥ω•α t | M I −ω∥ = 0 holds for any I ∈ I and ω ∈ (M I ) * .Let Z I denote the set of all minimal projections in the center Z(M I ) of M I and assume that Z I is a countable set.Moreover, we assume that M I = ℓ ∞ -z∈Z I zM I and zM I ∼ = B(h z ) for some finite dimensional Hilbert space h z .We particularly set M ∅ = C1.By [36,Lemma 7.1], the action α fixes elements in Z(M I ), and hence for any z ∈ Z I the restriction α| zM I gives a continuous flow on zM I .Now we fix an inverse temperature β ∈ R, and then there is a unique normal (α| zM I , β)-KMS state χ z on zM I .Remark 2.1.We later deal with a quasi-local algebra given by the inductive system of the unitary groups U(I) on ℓ 2 (I) ∼ = C |I| , and then M I are given as the group W * -algebras W * (U(I)) (and α is trivial).This setting might seem slightly strange at a glance for the readers who are familiar with the asymptotic representation theory.In the asymptotic representation theory of U(∞), we usually do not discriminate between U(I) and the unitary group U (|I|) of rank |I|.However, distinguishing them play an important role in the paper, and hence we deal with a family of W * -algebras indexed by I.
For any I ∈ I, we define a faithful normal conditional expectation E I : M I → Z(M I ) by Lemma 2.2.Let I, J ∈ I be disjoint and I ⊔ J ∈ I.For any x ∈ M I and y ∈ M J , we have Proof .Let x ∈ M I and y ∈ M J .Since every element in M I is a linear combination of (four) positive elements in M I (see, e.g., [2, Proposition II.3.12 (vi)]), we may assume that x is positive.For any z ∈ Z I⊔J with zx ̸ = 0, we define a state χ z,x on M J by χ z,x (y) := χ z (zxy)/χ z (zx) for any y ∈ M J .Since x ∈ (M J ) ′ , it is easy to check χ z,x is a normal (α| M J , β)-KMS state on M J .By [36,Lemma 7.3], we have χ z,x • E J = χ z,x , i.e., χ z (zxE J (y)) = χ z (zxy) for any z ∈ Z I⊔J .Thus, we have Replacing the roles of x and y, we have We recall that a state χ on M is said to be locally normal if χ| M I are normal on M I for any I ∈ I.Moreover, we define the following: Let us recall that x ∈ M is said to be analytic (for α) if the function t ∈ R → α t (x) ∈ M can be extended to an entire function on C. A state χ on M is called a (α, β)-KMS (Kubo-Martin-Schwinger ) state if χ(xα iβ (y)) = χ(yx) holds for any analytic elements x, y ∈ M. We remark that (α, β)-KMS states are nothing but tracial states when α is trivial.
The following is the first main result in the paper: Theorem 2.4.Let χ be a locally normal (α, β)-KMS state on M. The following are equivalent: (1) χ is multiplicative, (2) for any I ⊔ J ∈ I and z ∈ Z(M I ), w ∈ Z(M J ), we have χ(zw) = χ(z)χ(w).
Proof .Clearly, (1) implies (2).We assume that the condition (2) holds.Let I, J ∈ I be disjoint and x ∈ M I , y ∈ M J .We may assume that I ⊔ J ∈ I by replacing I or J by bigger intervals if necessary.Then we have (by [36,Lemma 7.3]).

Quasi-local algebras with shift translations
Here we assume the following additional setup: For every pair I, J ∈ I, if |I| = |J|, then there is a normal * -isomorphism γ I,J : M J → M I such that (1) γ I,I = id and γ I,J • γ J,K = γ I,K for any I, J, K ∈ I with See Section 3 for a concrete example of this setting.
Lemma 2.5.For any I, J ∈ I with |I| = |J|, we have Proof .For any z ∈ Z J we have χ γ I,J (z) •γ I,J | zM J = χ z by the uniqueness of a normal (α| zM J , β)-KMS states on zM J .Thus, for any x ∈ M J we have A state χ is said to be shift invariant if χ • γ I,J = χ| M J on M J for any I, J ∈ I with |I| = |J|.The aim here is to give a characterization of multiplicative shift invariant locally normal (α, β)-KMS states.
For any z I ∈ Z(M I ) and z J ∈ Z(M J ), we define z I • z J := E I∨J (z I γ J+j,J (z J )), where I ∨ J := I ⊔ (J + j) and j ∈ Z such that max I + 1 = min J + j.Namely, we have I ∨ J ∈ I.This operation defines an associative multiplication on Σ.Let z * ∈ Z(M * ) for each * = I, J, K ∈ I.For some j, k ∈ Z, we have (I ∨ J) ∨ K = I ∨ (J ∨ K) = I ⊔ (J + j) ⊔ (K + k).By Lemma 2.2, we have Thus, Σ becomes a unital algebra over C with unit 1 ∅ ∈ M ∅ = C1 ∅ , and Σ is a subalgebra of Σ.
Remark 2.7.As we mentioned in Remark 2.1, unlike the usual discussion of asymptotic representation theory, we deal with W * -algebras M I labeled by any finite intervals I in N or Z in the paper.Hence it seems natural to introduce Σ rather than Σ.However, if a quasi-local algebra M has shift translations, then M I depends only on |I|, and one can expect Σ to be sufficient to study appropriate (shift invariant) states.See Theorem 2.9.The unital algebra Σ is defined similarly to representation rings of Olshanski [27] and Ueda [35].If we deal with intervals of N, there exists an ideal of Σ such that the quotient should be understood as a dimension group of Vershik and Kerov (see [20,40,41,42] and also [35,Remark 3.4.1]).
We define a linear map γ : Σ → Σ by γ(z) := γ I+1,I (z) for any z ∈ Z(M I ) ⊂ Σ. Namely, we have γ we have Then ∼ is an equivalence relation on Σ, and [z] denotes the equivalence class of z ∈ Σ.By the above two equations, z, w ∈ Σ is a well-defined multiplication on Σ/∼, that is, Σ/∼ is also a unital algebra.Then we have following: Proposition 2.8.Two algebras Σ and Σ/∼ is isomorphic.
A certain class of KMS-states with respect to this R-flow relates to quantized characters of the infinite-dimensional quantum unitary group, which is a fundamental concept in the asymptotic representation theory for quantum groups (see [31,32]).Using Theorem 2.4, we show that some quantized characters dealt in [30] give multiplicative states on the quasi-local algebra (see Proposition 3.7).

Compact quantum groups
We first recall basic facts of compact quantum groups.See [21,24] for more details.
Let G be a compact quantum group, that is, G is a pair of unital C * -algebra C(G) and unital where ⊗ denotes the operation of minimal tensor product of C * -algebras.For a finite dimensional vector space V an invertible element , where we use the leg numbering notation for U ij .For any φ ∈ End(V ) * we call (φ ⊗ id)(U ) ∈ C(G) the matrix coefficient of U .It is known that the linear span of all matrix coefficients of finite dimensional corepresentations of G, denote by C[G], has a Hopf *algebra structure.Unlike ordinary compact groups, the antipode S G of C[G] generally does not satisfy S 2 G = id.In fact, for any finite dimensional corepresentation U on V there exists a positive invertible element We assume that ρ U also satisfies that Tr V (ρ U •) = Tr ρ −1 U • on the space of intertwining operators from U to itself.Under this assumption, ρ U is uniquely determined, and the scaling automorphism group τ urally has a unital * -algebra structure.For any corepresentation U on V , we obtain a *representation π U : U(G) → End(V ) by π U (x) := (id ⊗ x)(U ).Let G denote the set of all equivalence classes of irreducible corepresentations of G and fix representatives U λ for any λ ∈ G.
By definition, W * (G) has a W * -algebra structure and called the group W * -algebra of G.We define the dual of the scaling automorphism group τ for any t ∈ R and denote by the same symbol τ G the restriction of τ G to W * (G).In [31,32], we introduced the following notion, which is a fundamental concept in the asymptotic representation theory for quantum groups.

Quantum unitary groups
Now we discuss the quantum unitary groups.We fix a quantization parameter q ∈ (0, 1).Let I be the set of all finite intervals in N or Z and I ∈ I.We define a universal unital * -algebra C[U q (I)] generated by u ij (i, j ∈ I) and d −1 q,I subject to the relations where for any )] has a Hopf * -algebra structure and becomes a CQG algebra.Thus, by [21, Proposition 11.32], there exists a compact quantum group, denote by U q (I), such that its Hopf *algebra of matrix coefficients of finite dimensional corepresentations is isomorphic to C[U q (I)], and its C * -algebra C(U q (I)) is nothing but the universal C * -algebra generated by C[U q (I)].We call U q (I) the quantum unitary group.
Let U(I) be the group of unitary operators on ℓ 2 (I) ∼ = C |I| .It is well known that U q (I) and U(I) have the same representation theory.In particular, all irreducible (co-)representations of U q (I) and U(I) are labeled by S I := λ = (λ i ) i∈I ∈ Z I | λ is nonincreasing (see [21,25]).Thus, the group W * -algebra W * (U q (I)) is * -isomorphic to ℓ ∞ -λ∈S I End(V λ ), where V λ is the representation space of the irreducible corepresentations of U q (I) labeled by λ, and V λ has the same dimension of the irreducible representation of U(I) labeled by λ.Hence W * (U q (I)) is * -isomorphic to W * (U(I)), but the dual scaling automorphism group τ Uq(I) : R ↷ W * (U q (I)) is nontrivial unlike in the case of W * (U(I)).Moreover, we can describe τ Uq(I) explicitly using the representation theory of the quantized universal enveloping algebra U q gl I .
The quantized universal enveloping algebra U q gl I is a universal unital algebra generated by K i , K −1 i (i ∈ I) and E j , F j (j ∈ I\{max I}) subject to the relations Here we follow the notation in [21].It is known that U q gl I has a Hopf * -algebra structure, and there exists a non-degenerate dual pairing between two Hopf * -algebras U q gl I and C[U q (I)] (see [21,Theorem 9.18,Corollary 11.54]).Using this dual pairing, we can identify U q gl I with a * -subalgebra of U(U q (I)), and hence for any corepresentation U of U q (I) on V we obtain R. Sato a * -representation π U : U q gl I → End(V ), which is the restriction of π U : U(U q (I)) → End(V ) to U q gl I .Let U λ be an irreducible corepresentation of U q (I) labeled by λ ∈ S I .Then (π U λ , V λ ) is a highest weight representation with highest weight λ (see [21,Proposition 11.50]), and π U λ is hereinafter referred to as π λ .Moreover, we have ρ . We also have , where s λ is the Schur (Laurent) polynomial labeled by λ.

The quasi-local algebra from quantum unitary groups
Here we construct a quasi-local algebra from the quantum unitary groups U q (I).First, we check that the group W * -algebras W * (U q (I)) form an inductive system, and then we take their C * -inductive limit M(U q ).In Proposition 3.3, we show that the pair (M(U q ), (W * (U q (I)) I∈I )) is a quasi-local algebra.
Let I, J ∈ I such that I ⊆ J.By definition, there is a surjective unital * -homomorphism and δ Uq(I) • θ J I = θ J I ⊗ θ J I • δ Uq(J) holds true.Namely, U q (I) is an algebraic quantum subgroup of U q (J).By [24, Theorem 2.7.10],U q (I) is co-amenable, and hence U q (I) is a quantum subgroup of U q (J) (see [34,Lemma 2.7]).Namely, θ J I extends to a surjective unital * -homomorphism from C(U q (J)) to C(U q (I)).The dual map Θ J I : U(U q (I)) → U(U q (J)) given by Θ J I (x) := x • θ J I is an injective unital * -homomorphism, and we have See, e.g., [31] for more details.If I ⊆ J ⊆ K, then Θ K J • Θ J I = Θ K I holds true.Thus, the group W * -algebras W * (U q (I)) form an inductive system, and M(U q ) denotes their C * -inductive limit.We remark that the canonical * -homomorphism W * (U q (I)) → M(U q ) is injective since Θ J I is injective for any I, J ∈ I with I ⊆ J. Thus, we can freely identify W * (U q (I)) with a subalgebra of M(U q ).By the universality of M(U q ), we obtain a unique flow τ : R ↷ M(U q ) such that τt | W * (Uq(I)) = τ Uq(I) t for any t ∈ R and I ∈ I.For any I, J ∈ I, the tensor product C[U q (I)] ⊗ C[U q (J)] is also a CQG algebra, and hence we obtain the associated compact quantum group, denoted by U q (I) × U q (J).We have and hence W * (U q (I) × U q (J)) ∼ = W * (U q (I)) ⊗W * (U q (J)) taking the bounded parts of the above two * -algebras, where ⊗ denotes the operation of tensor products of W * -algebras (see, e.g., [2, Section III.1.5]).Lemma 3.2.Let I, J ∈ I.If I ∩ J = ∅, then W * (U q (I)) commutes with W * (U q (J)).Namely, (M(U q ), (U q (I)) I∈I ) satisfies condition (ql4).
Proof .The first two conditions (ql1), (ql2) are clear.Since Θ J I is unital for any I, J ∈ I with I ⊆ J, any W * (U q (I)) have a common identity 1 ∈ M, that is, (ql3) holds.The condition (ql4) follows from Lemma 3.2.■ The quasi-local algebra (M(U q ), (W * (U q (I))) I∈I ) also admits shift transformations.
Proof .We remark that there is k ∈ Z such that I = J + k.By definition, we obtain a unital *isomorphism r I,J : C[U q (I)] → C[U q (J)] by r I,J (u i+k,j+k ) := u ij (i, j ∈ J) and r I,J d −1 q,I = d −1 q,J , and we have where r J,I := r −1 I,J .Thus, U q (I) and U q (J) can be regarded as a quantum subgroup of each other.Therefore, there exists an injective normal * -isomorphism γ I,J : W * (U q (J)) → W * (U q (I)) satisfying (γ3), that is, τ Uq(I)t • γ I,J = γ I,J • τ Uq(J) t for any t ∈ R. Let I, J, K ∈ I.
For any ν ∈ S I⊔J the restriction of (π ν , V ν ) to U q gl I ⊗ U q gl J decompose into the direct sum where the multiplicity c ν λ,µ is the same in the classical case and determined by See, e.g., [33,Proposition 5.4].Therefore, for any λ ∈ S I and µ ∈ S J we have where |λ|

Examples of multiplicative states on M(U q )
Here we give examples of multiplicative states on the quasi-local algebra from the quantum unitary groups U q (I).In this section, let I be the set of all finite intervals of N, We first recall some facts about characters of U(∞) = lim − →I∈I U(I).We endow U(∞) with the inductive limit topology, that is, a function f on U(∞) is continuous if and only if f | U(I) is continuous on U(I) for any and f (e) = 1, where e ∈ U(∞) is the identity element.By definition, the set of all characters of U(∞) is a convex set, and it is known that every extreme character f of U(∞) is multiplicative, that is, f (uv) = f (u)f (v) for any u ∈ U(I) and v ∈ U (J) if I, J ∈ I are disjoint (see [43]).Moreover, it is known the following explicit description of the extreme characters of U(∞) (see [6,9,26,28,39,44]): There exists a bijection between the set of all extreme characters of U(∞) and the set Ω consisting of More precisely, for any ω ∈ Ω the corresponding extreme character f ω is given as where z runs over all eigenvalues of u and .
Remark 3.6.The W * -inductive limit of (W * (U q (I))) I∈I naturally has a structure like a Woronowicz algebra, and hence we can regard this as an inductive limit quantum group, called the infinite-dimensional quantum unitary group (see [32]).The R-flow τ extends to the W *inductive limit, and a normal (τ , −1)-KMS state is called a quantized character.Moreover, there exists a bijection between the quantized characters and the locally normal (τ , −1)-KMS states on M(U q ).Proposition 3.7.For any q ∈ (0, 1] and ω ∈ Ω q , the corresponding state χ (q) ω on M(U q ) is multiplicative.
Proof .By Theorem 2.4, it suffices to show that χ On the other hand, we have i∈I⊔J Therefore, we have χ The restriction of χ (q) ω to W * (U q (I)) is a quantized character of U q (I).By equation (3.3), it is multiplicative on the "maximal tori" of U q (I).By Proposition 3.7, the state χ (q) ω , which gives a quantized characters having a multiplicative form on the maximal tori, is really multiplicative in the sense of Definition 2.3.In the classical case (q = 1), the multiplicative characters are extreme characters.On the other hand, most of χ (q) ω is not extreme quantized character if 0 < q < 1.Thus, the χ (q) ω and Proposition 3.7 seem to indicate an essential difference in the character theory of U(∞) and U q (∞).

Quasi-local algebras and Gaussian fluctuation limits
In this section, we mention a central limit theorem with respect to multiplicative states on quasi-local algebras (see Theorem 4.1), and its proof is given in Appendix A. We mainly refer to the book by Petz [29] about Weyl CCR algebras and quasi-free states.
Throughout this section and Appendix A, let I be the set of all finite intervals in Z and (M, (M I ) I∈I ) a quasi-local algebra.Moreover, we follow the same setup in Section 2. Namely, M has an action α : R ↷ M such that α t (M I ) = M I and lim t→0 ∥ω•α t −ω∥ = 0 hold for any I ∈ I and ω ∈ (M I ) * .In addition, we suppose that M admits a shift translations.We remark that this assumption is equivalent to the existence of action γ : Z ↷ M such that γ commutes with α and γ j (M I ) = M I+j holds for any I ∈ I and j ∈ Z.Clearly, γ j preserves M 0 := I∈I M I for every j ∈ Z.Throughout this section, we fix a γ-invariant multiplicative state χ on M.Moreover, we fix an increasing sequence (I n ) ∞ n=1 in I such that ∞ n=1 I n = Z.For any x ∈ M and I ∈ I the local fluctuation F I (x) of x on I is defined by Let F n (x) := F In (x) for every n = 1, 2, . . . .Since χ(xγ j (y)) = χ(x)χ(y) if |j| is sufficiently large, we have j∈Z |χ(xγ j (y)) − χ(x)χ(y)| < ∞.Thus, the real vector space M 0,sa has the symmetric bilinear form s χ and the symplectic form σ χ given by for any x, y ∈ M 0,sa .We denote by W χ the Weyl CCR algebra associated with the symplectic space (M 0,sa , σ χ ), that is, W χ is the universal C * -algebra generated by w(x) (x ∈ M 0,sa ) subject to the relations w(x) * = w(−x), w(x)w(y) = e σχ(x,y) w(x + y), x, y ∈ M 0,sa .
The following is the main theorem in this section and the proof is given in Appendix A.
We now apply Theorem 4.1 to multiplicative characters of U(∞).We recall that M(U ) is the C * -inductive limit of the group W * -algebras W * (U(I)).Let M(U ) 0 := I∈I W * (U(I)) and M(U ) 0,sa the space of all self-adjoint elements in M(U ) 0 .For any ω ∈ Ω, we denote by χ ω the locally normal tracial state on M(U ) satisfying equation (3.3) with q = 1.By Proposition 3.7, χ ω is multiplicative on M(U ).
By equation (3.3) (see also [30,Lemma 7.1]), only if q = 1, then χ ω is γ-invariant, and hence the corresponding symmetric bilinear form s χω and the symplectic form σ χω are well defined.Let W χω denote the Weyl CCR algebra associated with (M(U ) 0,sa , σ χω ) and φ χω the quasi-free state on W χω associated with s χω .Corollary 4.3.For any k ∈ N and x 1 , . . ., x k ∈ M(U ) 0,sa , we have By this corollary, the extreme (i.e., multiplicative) characters of U(∞) produce Gaussian fluctuation limits on the group W * -algebras W * (U(I)).Moreover, since χ ω is tracial, we have s χω ≡ 0, and hence the Weyl CCR algebra W χω is commutative.Namely, we obtain a commutative Gaussian family as limits of F n (x 1 ), . . ., F n (x k ) for any x 1 , . . ., x k ∈ M(U ) 0,sa .However, to the best of the author's knowledge, a probabilistic description of them is unclear.

A Proof of Theorem 4.1
Here we show Theorem 4.1.The strategy is essentially the same as [14,16], and hence the proof might be clear for the experts.However, we give this appendix for the reader's convenience.We use the same notations in the previous section.
For any n = 1, 2, . . ., we set and take a partition mn+1 lie in order from left to right (see Figure 1).We remark that L (n) mn+1 < p n + q n .For x ∈ M, we define where l .We remark that F n (x) = s J n (x) + s L n (x).
Lemma A.1.Let x, y, z ∈ M 0 .The following hold true: (1) lim n→∞ s J n (x), s L n (y) = 0, lim n→∞ s J n (x), Proof .For the claim (1), we show only the first case, and the other statements can be proved by similar ways.By definition, p n /|I n |, q n /|I n |, and q n m n /|I n | converge to 0 as n → ∞.Thus, we have where we remark that k∈Z ∥[γ k (x), y]∥ < ∞ since x, y ∈ M 0 .We show the claim (2).By the Jacobi identity, it suffices to show that s J n (y), s J n (z), s L n (x) converges to 0 as n → ∞, and we have where we remark that k We obtain the following Gaussian fluctuation limit of single operator.
Proof .We may assume that χ(x) = 0. Using the formulas e i(a+b) −e ia = where we remark that χ s L n (x)e i(1−t)s J n (x) e itFn(x) ≤ χ s L n (x) 2 1/2 by the Cauchy-Schwarz inequality.By Lemma A.1, s J n (x), s L n (x) converges to 0 as n → ∞.We also have as n → ∞, where k∈Z |χ(xγ k (x))| < ∞ since χ(x) = 0 and χ is multiplicative.Thus, we have Therefore, it suffices to show that lim n→∞ χ(e is J n (x) ) = e −sχ(x,x)/2 .Since x ∈ M 0 , for sufficiently large n, the intervals L (n) l are large enough and {F J (n) l (x)} mn l=1 commute mutually.Since χ is γ-invariant and multiplicative, we have .
By the Taylor expansion theorem, there exists t ∈ [0, 1] such that where we remark that and lim n→∞ χ F J as n → ∞. ■ For any x, y ∈ M, we define L(x, y) := e ix e iy − e i(x+y) e − 1 2 [x,y] .

Figure 1 .
Figure 1.Partition of I n .
σ) := the number of inversions in σ.The above defining relations are the same in the book [21, Sections 9.2.1, 9.2.3, and 9.2.4], but we use the different symbol C[U q To show them, we use the following inequalities (see, e.g., [22, proof of Lemma 2.2], [13, proof of Theorem 4.1]): ia , e ib ≤ a, e ib ≤ ∥[a, b]∥ for any self-adjoint elements a, b of a C * -algebra.By Lemma A.1, we haveA n ≤ e iFn(x) − e is J n (x) e is L n (x) + e iFn(y) − e is J n (y) e is L n (y) + [e is L n (x) , e is J n (y) ]+ e is J n (x+y) e is L n (x+y) − e iFn(x+y) e n (y), s L n (y) + s L n (x), s J n (y) → 0 as n → ∞,andC n ≤ e − 1 2 [s J n (x),s J n (y)] , e is L n (x+y) e 1 2 ∥[s L n (x),s L n (y)]∥