PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 6, June 2010, Pages 2113–2123 S 0002-9939(10)10270-6 Article electronically published on February 9, 2010

TOEPLITZ-COMPOSITION C∗-ALGEBRAS FOR CERTAIN FINITE BLASCHKE PRODUCTS

HIROYASU HAMADA AND YASUO WATATANI

(Communicated by Marius Junge)

Abstract. Let R be a finite Blaschke product of degree at least two with R(0) = 0. Then there exists a relation between the associated composition ∗ operator CR on the and the C -algebra OR(JR) associated with ◦n the complex (R )n on the Julia set JR. We study the ∗ C -algebra TCR generated by both the CR and the Toeplitz operator Tz to show that the quotient algebra by the ideal of the ∗ compact operators is isomorphic to the C -algebra OR(JR), which is simple and purely infinite.

1. Introduction Let D be the open unit disk in the complex plane and H2(D)betheHardy (Hilbert) space of analytic functions whose power series have square-summable coefficients. For an analytic self-map ϕ : D → D, the composition operator Cϕ : 2 2 2 H (D) → H (D) is defined by Cϕ(g)=g ◦ ϕ for g ∈ H (D) and is known to be a by the Littlewood subordination theorem [14]. The study of composition operators on the Hardy space H2(D) gives a fruitful interplay between complex analysis and operator theory as shown, for example, in the books of Shapiro [29], Cowen and MacCluer [4] and Mart´ınez-Avenda˜no and Rosenthal [15]. Since the work by Cowen [2], good representations of adjoints of composition operators have been investigated. Consult Cowen and Gallardo-Guti´errez [3], Mart´ın and Vukoti´e [17], Hammond, Moorhouse and Robbins [8], and Bourdon and Shapiro [1] to see recent achievements in adjoints of composition operators with rational symbols. In this paper, we only need an old result by McDonald in [18] for finite Blaschke products. On the other hand, for a branched covering π : M → M, Deaconu and Muhly [6] introduced a C∗-algebra C∗(M,π)astheC∗-algebra of the r-discrete groupoid constructed by Renault [27]. In particular, they study rational functions R on the ◦n Riemann sphere Cˆ. Iterations (R )n of R by composition give complex dynam- ical systems. In [11] Kajiwara and the second-named author introduced slightly

Received by the editors October 23, 2008, and, in revised form, October 8, 2009. 2010 Subject Classification. Primary 46L55, 47B33; Secondary 37F10, 46L08. Key words and phrases. Composition operator, Blaschke product, Toeplitz operator, C∗- algebra, complex dynamical system.

c 2010 American Mathematical Society Reverts to public domain 28 years from publication 2113

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∗ different C -algebras OR(Cˆ), OR(JR)andOR(FR), associated with the complex ◦n dynamical system (R )n on the Riemann sphere Cˆ, the Julia set JR and the Fatou ∗ set FR of R.TheC -algebra OR(JR)isdefinedasaCuntz-Pimsneralgebra[26] ∗ | of a Hilbert bimodule, called a C -correspondence, C(graph R JR )overC(JR). We regard the algebra OR(JR) as a certain analog of the crossed product C(ΛΓ) Γof C(ΛΓ) by a boundary action of a Kleinian group Γ on the limit set ΛΓ. The aim of this paper is to show that there exists a relation between composition ∗ operators on the Hardy space and the C -algebras OR(JR) associated with the ◦n complex dynamical systems (R )n on the Julia sets JR. ∗ We recall that the C -algebra T generated by the Toeplitz operator Tz con- tains all continuous symbol Toeplitz operators, and its quotient by the ideal of the compact operators on H2(T) is isomorphic to the commutative C∗-algebra C(T) of all continuous functions on T. For an analytic self-map ϕ : D → D,wedenote ∗ by TCϕ the Toeplitz-compostion C -algebra generated by both the composition operator Cϕ and the Toeplitz operator Tz. Its quotient algebra by the ideal K of the compact operators is denoted by OCϕ. Recently Kriete, MacCluer and Moor- ∗ house [12, 13] studied the Toeplitz-compostion C -algebra TCϕ for a certain linear ∗ fractional self-map ϕ. They describe the quotient C -algebra OCϕ concretely as a −2πiθ subalgebra of C(Λ) ⊗ M2(C) for a compact space Λ. If ϕ(z)=e z for some ∗ irrational number θ, then the Toeplitz-composition C -algebra TCϕ is an exten- sion of the irrational rotation algebra Aθ by K and studied by Park [25]. Jury [9, 10] investigated the C∗-algebra generated by a group of composition operators with the symbols belonging to a non-elementary Fuchsian group Γ to relate it with extensions of the crossed product C(T) ΓbyK. In this paper we study the class of finite Blaschke products R of degree n ≥ 2 with R(0) = 0. The boundary T of the open unit disk D is the Julia set JR of the Blaschke product R. We show that the quotient algebra OCR of the Toelitz- ∗ ∗ composion C -algebra TCR by the ideal K is isomorphic to the C -algebra OR(JR) ◦n associated with the complex dynamical system (R )n, which is simple and purely infinite. We should remark that the notion of transfer operator considered by Exel in [7] is one of the keys to clarifying the above relation. In fact the corresponding operator of the composition operator in the quotient algebra is the implementing isometry operator. The Toeplitz-composition C∗-algebra depends on the analytic structure of the Hardy space by construction. The finite Blaschke product R is not conjugate with zn by any M¨obius automorphism unless R(z)=λzn. But we can show that the quotient algebra OCR is isomorphic to OCzn as a corollary of our main theorem. This enables us to compute K0(OCR)andK1(OCR) easily.

2. Toeplitz-composition C∗-algebras Let L2(T) denote the square integrable measurable functions on T with respect to the normalized Lebesgue measure. The Hardy space H2(T) is the closed subspace of L2(T) consisting of the functions whose negative Fourier coefficients vanish. We put H∞(T):=H2(T) ∩ L∞(T). 2 The Hardy space H (D) is the consisting of all analytic functions ∞ k D ∞ | |2 ∞ g(z)= k=0 ckz on the open unit disk such that k=0 ck < . The inner

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product is given by ∞ (g|h)= ckdk k=0 ∞ k ∞ k for g(z)= k=0 ckz and h(z)= k=0 dkz . We identify H2(D) with H2(T) by a unitary U : H2(D) → H2(T). We note that g = Ug is given as

g(eiθ) := lim g(reiθ)a. e.θ

r→1− for g ∈ H2(D) by Fatou’s theorem. Moreover the inverse f = U ∗f is given as a

Poisson integral 2π iθ 1 it f(re ):= Pr(θ − t)f(e )dt 2π 0 2 for f ∈ H (T), where Pr is the Poisson kernel defined by 1 − r2 P (θ)= , 0 ≤ r<1, 0 ≤ θ ≤ 2π. r 1 − 2r cos θ + r2 2 2 2 ∞ Let PH2 : L (T) → H (T) ⊂ L (T) be the projection. For a ∈ L (T), the 2 2 Toeplitz operator Ta on H (T) is defined by Taf = PH2 af for f ∈ H (T). Let ϕ : D → D be an analytic self-map. Then the composition operator Cϕ on 2 2 H (D) is defined by Cϕg = g ◦ ϕ for g ∈ H (D). By the Littlewood subordination

theorem, Cϕ is always bounded. We can regard Toeplitz operators and composition operators as acting on the ∗ same Hilbert space by the unitary U above. More precisely, we put Ta = U TaU ∗ and Cϕ = UCϕU .Ifϕ is an inner function, then we know that g◦ ϕ = g ◦ ϕ 2 for g ∈ H (D) by Ryff [28, Theorem 2]. Therefore we may write Cϕf = f ◦ ϕ for f ∈ H2(T).

∗ Definition. For an analytic self-map ϕ :D → D,wedenotebyTCϕ the C -

algebra generated by the Toeplitz operator Tz and the composition operator Cϕ on 2 ∗ ∗ H (D). The C -algebra TCϕ is called the Toeplitz-composition C -algebra with 2 symbol ϕ.SinceTCϕ contains the ideal K(H (D)) of compact operators, we define ∗ ∗ 2 a C -algebra OCϕ to be the quotient C -algebra TCϕ/K(H (D)). By the unitary 2 2 ∗ U : H (D) → H (T) above, we usually identify TCϕ with the C -algebra generated by Tz and Cϕ. We also use the same notation TCϕ and OCϕ for the operators on

2 H (T). But we sometimes need to treat them carefully. Therefore we often use the notation g = Ug and f = U ∗f to avoid confusion in the paper. Wise readers may neglect this troublesome notation.

3. C∗-algebras associated with complex dynamical systems We recall the construction of Cuntz-Pimsner algebras [26]. Let A be a C∗- algebra and X be a Hilbert right A-module. We denote by L(X) the algebra of the adjointable bounded operators on X.Forξ, η ∈ X, the operator θξ,η is defined by θξ,η(ζ)=ξ(η|ζ)A for ζ ∈ X. The closure of the linear span of these operators is denoted by K(X). We say that X is a Hilbert C∗-bimodule (or C∗-correspondence) over A if X is a Hilbert right A-module with a *-homomorphism φ: A → L(X). ∞ ⊗n We always assume that X is full and φ is injective. Let F (X)= n=0 X be

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the full Fock module of X with a convention X⊗0 = A.Forξ ∈ X, the creation operator Tξ ∈ L(F (X)) is defined by

Tξ(a)=ξa and Tξ(ξ1 ⊗···⊗ξn)=ξ ⊗ ξ1 ⊗···⊗ξn.

We define iF (X) : A → L(F (X)) by

iF (X)(a)(b)=ab and iF (X)(a)(ξ1 ⊗···⊗ξn)=φ(a)ξ1 ⊗···⊗ξn

∗ for a, b ∈ A. The Cuntz-Toeplitz algebra TX is the C -algebra acting on F (X) generated by iF (X)(a) with a ∈ A and Tξ with ξ ∈ X. →T ∗ Let jK : K(X) X be the homomorphism defined by jK (θξ,η)=TξTη .We −1 consider the ideal IX := φ (K(X)) of A.LetJX be the ideal of TX generated by {iF (X)(a)−(jK ◦φ)(a); a ∈ IX }. Then the Cuntz-Pimsner algebra OX is defined as the quotient TX /JX .Letπ : TX →OX be the quotient map. We set Sξ = π(Tξ) and i(a)=π(iF (X)(a)). Let iK : K(X) →OX be the homomorphism defined by ∗ ◦ ◦ ∈ iK (θξ,η)=SξSη .Thenπ((jK φ)(a)) = (iK φ)(a)fora IX . ∗ The Cuntz-Pimsner algebra OX is the universal C -algebra generated by i(a) with a ∈ A and Sξ with ξ ∈ X, satisfying the fact that i(a)Sξ = Sφ(a)ξ, Sξi(a)= ∗ | ∈ ∈ ◦ ∈ Sξa, Sξ Sη = i((ξ η)A)fora A, ξ,η X and i(a)=(iK φ)(a)fora IX .We { }n usually identify i(a) with a in A.IfA is unital and X has a finite basis ui i=1 in n | the sense that ξ = i=1 ui(ui ξ)A, then the last condition can be replaced by the { }m ⊂ m ∗ fact that there exists a finite set vj j=1 X such that j=1 Svj Svj = I.Then { }m vj j=1 becomes another finite basis of X automatically. Next we introduce the C∗-algebras associated with complex dynamical systems ◦n as in [11]. Let R be a rational function of degree at least two. The sequence (R )n of iterations of composition by R gives a complex dynamical system on the Riemann sphere Cˆ = C ∪{∞}.TheFatousetFR of R is the maximal open subset of Cˆ on ◦n which (R )n is equicontinuous (or a normal family) and the Julia set JR of R is the complement of the Fatou set in Cˆ.Wedenotebye(z0) the branch index of R at z0.LetA = C(Cˆ)andX = C(graph R) be the set of continuous functions on Cˆ and graph R respectively, where graph R = {(x, y) ∈ Cˆ 2 ; y = R(x)} is the graph of R.ThenX is an A-A bimodule by

(a · ξ · b)(x, y)=a(x)ξ(x, y)b(y),a,b∈ A, ξ ∈ X.

We define an A-valued inner product ( | )A on X by (ξ|η)A(y)= e(x)ξ(x, y)η(x, y),ξ,η∈ X, y ∈ Cˆ. x∈R−1(y)

Thanks to the branch index e(x), the inner product above gives a continuous func- tion and X is a full Hilbert bimodule over A without completion. The left action of A is unital and faithful. Since the Julia set JR is completely invariant under R, i.e., R(JR)=JR = −1 | → R (JR), we can consider the restriction R JR : JR JR, which will often be | { ∈ × } denoted by the same letter R. Let graph R JR = (x, y) JR JR ; y = R(x) be | | the graph of the restriction map R JR and XR = C(graph R JR ). In the same way as above, XR is a full Hilbert bimodule over C(JR).

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∗ Definition. The C -algebra OR(Cˆ)onCˆ is defined as the Cuntz-Pimsner algebra of the Hilbert bimodule X = C(graph R)overA = C(Cˆ). We also define the C∗- algebra OR(JR) on the Julia set JR as the Cuntz-Pimsner algebra of the Hilbert | bimodule XR = C(graph R JR )overA = C(JR).

4. Transfer operators and adjoints of composition operators In the rest of this paper, we assume that R is a finite Blaschke product of degree at least two with R(0) = 0; that is, − − n1 z − z n1 z − z R(z)=λz k = λ k ,z∈ Cˆ, 1 − zkz 1 − zkz k=1 k=0

where n ≥ 2,z1,...,zn−1 ∈ D, |λ| =1andz0 =0.ThusR is a rational function with degree deg R = n.SinceR is an inner function, R is an analytic self-map on D. We consider the composition operator CR with symbol R.SinceR is inner and R(0) = 0, CR is an isometry by Nordgren [24]. We note the following fact: −  n1 −| |2 zR (z) 1 zk ∈ T (1) =1+ 2 > 0,z . R(z) |z − zk| k=1 Thus R has no branched points on T and the branch index e(z) = 1 for any z ∈ T. Furthermore the Julia set JR of R is T (see, for example, [19, pages 70-71]), and it coincides with the boundary of the disk D. Let A = C(T)andh ∈ A be a positive invertible element. Set graph R|T = 2 {(z,w) ∈ T ; w = R(z)} and XR,h = C(graph R|T). We define (a · ξ · b)(z,w)=a(z)ξ(z,w)b(w), (ξ|η)A,h(w)= h(z)ξ(z,w)η(z,w) z∈R−1(w)

for a, b ∈ A and ξ,η ∈ XR,h.WeseethatXR,h is a pre-Hilbert A-A bimodule whose left action is faithful. Since the Julia set JR = T has no branched point, for the constant function h =1,XR,1 has a finite basis and coincides with the Hilbert | { }N A-A bimodule XR = C(graph R T). Let uk k=1 be a basis of XR. For a positive ∈ −1/2 { }N invertible element h A, put vk = h uk.Then vk k=1 is a basis of XR,h,and XR,h is also a Hilbert module without completion. ∗ Definition. The C -algebra OR,h(T) is defined as the Cuntz-Pimsner algebra of the ∗ Hilbert bimodule XR,h = C(graph R|T)overA = C(T). The C -algebra OR,h(T) ∗ is the universal C -algebra generated by {Sˆξ ; ξ ∈ XR,h} and A satisfying the following relations: N ∗ ∗ aSˆ = Sˆ · , Sˆ b = Sˆ · , Sˆ Sˆ =(ξ|η) , Sˆ Sˆ = I ξ a ξ ξ ξ b ξ η A,h vk vk k=1 ∗ for a, b ∈ A and ξ,η ∈ XR,h.TheC -algebra OR,h(T) is in fact a topological quiver algebra in the sense of Muhly and Tomforde [21]. See also Muhly and Solel [20, Example 5.4].

We will use the symbol a and Sξ to denote the generator of OR(JR)fora ∈ A and ξ ∈ XR.

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In the rest of this paper, we choose and fix h ∈ A defined by nR(z) n h(z)= = ,z∈ T. zR(z) |R(z)| Then h is positive and invertible by (1). We need and collect several facts as lemmas to prove our main theorem. Some of them might be considered folklore. Lemma 1. Let R be a finite Blaschke product of degree at least two with R(0) = ∗ ∗ 0. Then the C -algebra OR,h(T) is isomorphic to the C -algebra OR(JR) by an 1/2 isomorphism Φ such that Φ(a)=a and Φ(Sˆξ)=h Sξ for a ∈ A, ξ ∈ XR,h. 1/2 Proof. Let Vξ = h Sξ for ξ ∈ XR,h. Then we have that N ∗ 1/2 1/2 ∗ aV = V · ,Vb = V · ,VV =(h · ξ|h · η) =(ξ|η) , V V = I ξ a ξ ξ ξ b ξ η A A,h vk vk k=1

for a, b ∈ A and ξ,η ∈ XR,h. By universality, we have the desired isomorphism. 

Definition. For a function f on T, we define a function LR(f)onT by 1 R(z) L (f)(w)= h(z)f(z)= f(z),w∈ T. R n zR(z) z∈R−1(w) z∈R−1(w) 2 We mainly consider the restrictions of LR to A = C(T)andH (T) and use the same notation if no confusion can arise. The notion of transfer operator by Exel in [7] is one of the keys to clarifying our situation. Lemma 2. Let R be a finite Blaschke product of degree at least two with R(0) = 0. Let A = C(T) and α : A → A be the unital *-endomorphism defined by (α(a))(z)= a(R(z)) for a ∈ A and z ∈ T. Then the restriction of LR to A = C(T) is a transfer operator for the pair (A, α) in the sense of Exel; that is, LR is a positive linear map such that LR(α(a)b)=aLR(b) for a, b ∈ A.

Moreover LR satisfies the fact that LR(1) = 1.

Proof. The only non-trivial statement is to show that LR(1) = 1. But this is an easy calculation as follows: We fix w ∈ T and let α1,...,αn be exactly different solutions of R(z)=w. By the partial fraction decomposition, there exists λ1,...,λn ∈ C such that R(z)/z n λ = k . R(z) − w z − αk k=1 →∞ n Multiplying by z and letting z ,wehave k=1 λk =1. Moreover multiplying R(αl) by z − αl and letting z → αl,wegetλl =  . This shows that LR(1) = 1. αlR (αl) See, for example, [5].  Let R be a finite Blaschke product of degree at least two with R(0) = 0. J. N. McDonald [18] calculated the adjoint of CR and gave a formula. We follow some of his argument, but we also need a different formula to prove our main theorem. We claim that there exist θ0 ∈ [0, 2π] and a strictly increasing continuously differ- entiable function ψ :[θ0 − 2π, θ0] → R such that ψ(θ0 − 2π)=0,ψ(θ0)=2nπ and R(eiθ)=eiψ(θ).

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Lemma 3. Let R be a finite Blaschke product of degree at least two with R(0) = 0. 2 2 2 2 Then LR(H (T)) ⊂ H (T) and the restriction LR : H (T) → H (T) is a bounded ∗ operator such that CR = LR. −1 Proof. Let ψ be the map defined above. Set t = ψ(θ)andσk(t)=ψ (t+2(k−1)π) for 1 ≤ k ≤ n and 0 ≤ t ≤ 2π. If we differentiate R(eiσk(t))=eit with respect to t, then iσ (t)  R(e k ) σk(t)=  . R (eiσk(t))eiσk(t) l Therefore for ξl(z):=z we have that 2π θ0 1 iθ l iθ 1 iθ l iθ (CRξl|f)= R(e ) f(e )dθ = R(e ) f(e )dθ 2π 2π − 0 θ0 2π 2nπ n 2π 1 −1 1  ilt iψ (t) −1  ilt iσk(t) = e f(e )(ψ (t)) dt = e f(e )σk(t)dt 2π 0 2π 0 k=1 2π 1 ilt it = e LR(f)(e )dt =(ξl|LR(f)) 2π 0 ∗ 2 for f ∈ H (T)andl ≥ 0. Thus CR = LR and LR is bounded.  We need an appropriate basis of H2(T)forR.Let ⎧ ⎪ −| |2 ⎪ 1 β0 ⎨⎪ ,l=0, 1 − β0z el(z)= − ⎪ −| |2 l 1 ⎪αl 1 βl z − βk ⎩⎪ ,l≥ 1, 1 − β z 1 − β z l k=0 k k ≥ ≤ ≤ − ∞ −| | where αkn+l = λ ,βkn+l = zl for k 0and0 l n 1. Since k=0(1 βk )= ∞ { }∞ 2 T , ek k=0 is an orthonormal basis of H ( ) as in Ninness, Hjalmarsson and Gustafsson [23, Theorem 2.1], and Ninness and Gustafsson [22, Theorem 1]. We write ⎧ ⎪ ⎨⎪1,l=0, 1 −|z |2 Q (z)= l and R (z)= l−1 l − l ⎪ z − zk 1 zlz ⎩⎪ ,l≥ 1. 1 − zkz k=0 k k Thus ekn+l = QlRlR for k ≥ 0and0≤ l ≤ n − 1, where R is the k-th power of R with respect to pointwise multiplication. Proposition 4. Let R be a finite Blaschke product of degree at least two with R(0) = 0. Then for any a ∈ C(T), we have ∗ CR TaCR = TLR(a). Proof. We first examine the case that a(z)=zj for j ≥ 0. Consider the L2- { }∞ expansion of a by the basis el l=0: ∞ l a = clR + g, l=0

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⊥ 2 m where g ∈ (Im CR) ∩ H (T). For ξm(z)=z with m ≥ 0, we have ∞ m l+m m (TaCRξm)(z)=a(z)R(z) = clR(z) + g(z)R(z) . l=0 It is clear that Im CR = span{ekn ; k ≥ 0} and

⊥ 2 (Im CR) ∩ H (T)=span{ekn+l ; k ≥ 0, 1 ≤ l ≤ n − 1},

m ⊥ where span means the closure of a linear span. Therefore gR is also in (Im CR) ∩ 2 H (T). Since CR is an isometry, we have that ∞ ∗ l+m (CR TaCRξm)(z)= clz . l=0 ∗ On the other hand, CR = LR by Lemma 3; hence ∞ l LR(a)= clLR(R ). l=0 By Lemma 2, 1 L (Rl)(w)= h(z)R(z)l =(L (1)(w))wl = wl. R n R z∈R−1(w) Thus ∞ l 2 LR(a)(w)= clw as an L -convergence. l=0 ∞ Since LR(a) ∈ H (T), we have ∞ L L l+m (TLR(a)ξm)(w)=( R(a)ξm)(w)=(Mξm R(a))(w)= clw l=0 ≥ for m 0, where Mξm is a multiplication operator by ξm. Therefore we obtain that ∗ CR TaCR = TL (a). R ∗ ≤ For the remaining case where j 0, the formula CR TaCR = TLR(a) also holds L ∗  because R is positive and Ta = Ta. Lemma 5. Let the notation be as above. Then n ∗ ∗ TQk−1Rk−1 CRCR (TQk−1Rk−1 ) = I. k=1 Proof. We have l TQk−1Rk−1 CRξl = Qk−1Rk−1R = eln+(k−1) ≤ ≤ ≥ for 1 k n and l 0. Thus TQk−1Rk−1 CR is an isometry, and the desired equality follows.  We can now state our main theorem.

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Theorem 6. Let R be a finite Blaschke product of degree at least two and let R(0) = 0; that is, − n1 z − z R(z)=λz k , 1 − zkz k=1 ∗ where n ≥ 2, |λ| =1and |zk| < 1. Then the quotient C -algebra OCR of the ∗ Toeplitz-composition C -algebra TCR by the ideal of compact operators is isomor- ∗ phic to the C -algebra OR(JR) associated with the complex dynamical system on ∈ T the Julia set JR by an isomorphism Ψ such that Ψ(π(Ta)) = a for a C( ) and R TC OC Ψ(π(CR)) = zR S1,whereπ is the canonical quotient map R to R and 1 ∗ is the constant map in XR taking constant value 1. Moreover the C -algebra OCR is simple and purely infinite.

Proof. For any ξ ∈ XR,h and a ∈ A,letp(z)=ξ(z,R(z)),ρ(a)=π(Ta),Vξ = 1/2 n ρ(p)π(CR). We have 1/2 ρ(a)Vξ = n ρ(ap)π(CR)=Va·ξ. Since, for b ∈ A π(CR)ρ(b)=π(CRTb)=π(Tb◦RCR)=ρ(b ◦ R)π(CR), we have that 1/2 1/2 Vξρ(b)=n ρ(p)π(CR)ρ(b)=n ρ(p)ρ(b ◦ R)π(CR) 1/2 = n ρ(p(b ◦ R))π(CR)=Vξ·b.

For any η ∈ XR,h, define q(z)=η(z,R(z)). By Proposition 4, ∗ ∗ ∗ Vξ Vη = nπ(CR )ρ(p)ρ(q)π(CR)=nπ(CR TpqCR) L | = nπ(TLR(pq))=ρ(n R(pq)) = ρ((ξ η)A,h). −1/2 Set vk(z,R(z)) = n Qk−1(z)Rk−1(z)for1≤ k ≤ n. By Lemma 5, n n ∗ V V ∗ = π(T C C (T )∗)=I. vk vk Qk−1Rk−1 R R Qk−1Rk−1 k=1 k=1

By the universality and the simplicity of OR,h(T), there exists an isomorphism Ω:OR,h(T) →OCR such that Ω(Sˆξ)=Vξ and Ω(a)=ρ(a)forξ ∈ XR,h,a∈ A. Let Φ be the map in Lemma 1 and put Ψ = Φ ◦ Ω−1. Then Ψ is the desired ∗ isomorphism. Since it is proved in [11] that the C -algebra OR(JR)issimpleand ∗ purely infinite, so is the C -algebra OCR. The rest is now clear.  Remark. It is important to notice that the element π(CR) of the composition op- erator in the quotient algebra corresponds exactly to the implementing isometry N operator in Exel’s crossed product A α,LR in [7], which depends on the transfer opearator LR. It follows directly from the fact that OR,h(T) is naturally isomorphic N to A α,LR . n Example. Let R(z)=z for n ≥ 2. Then the Hilbert bimodule XR over A = C(T) n √1 i−1 is isomorphic to A as a right A-module. In fact, let ui(z,w)= n z for i = | { } 1,...,n.Then(ui uj )A = δi,j I and u1,...,un is a basis of XR. Hence Si := Sui (i =1, ..., n) are generators of the Cuntz algebra On.Weseethat(z ·ui)(z,R(z)) = n ui+1(z,R(z)) for i =1, ..., n − 1and(z · un)(z,R(z)) = z =(u1 · z)(z,R(z))

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and that the left multiplication by z is a unitary U. Therefore the C∗-algebra OR(JR) associated with the complex dynamical system on the Julia set JR is the ∗ universal C -algebra generated by a unitary U and n isometries S1, ..., Sn satisfying ∗ ··· ∗ − S1S1 + + SnSn = I, USi = Si+1 for i =1, ..., n 1andUSn = S1U.Inthis way, the operator S1 corresponds to the element π(CR) ∈OCR of the composition operator CR and the unitary U corresponds to the element π(Tz) ∈OCR.Moreover n ∗ we find that the commutation relation U S1 = S1U in the C -algebra OR(JR)and ∗ the commutation relation TR(z)CR = CRTz in the Toeplitz-composition C -algebra TCR are essentially the same. The Toeplitz-composition C∗-algebra depends on the analytic structure of the Hardy space by the construction. The finite Blaschke product R is not conjugate with zn by any M¨obius automorphism unless R(z)=λzn. But we can show that the quotient algebra OCR is isomorphic to OCzn as a corollary of our main theorem. Corollary 7. Let R be a finite Blaschke product of degree n ≥ 2 with R(0) = 0. ∗ ∗ Then the quotient C -algebra OCR is isomorphic to the C -algebra OCzn . Moreover K0(OCR) Z ⊕ Z/(n − 1)Z and K1(OCR) Z. Proof. Let ψ be the strictly increasing continously differentiable function defined in the paragraph before Lemma 3. Since

iθ  iθ n−1 2  e R (e ) 1 −|zk| ψ (θ)= iθ =1+ iθ 2 > 1, R(e ) |e − zk| k=1 n R|T is an expanding map of degree n and R|T is topologically conjugate to z on T by [30]. See a general condition given in Martin [16]. Therefore the statement follows from the above theorem and the results in [11]. 

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Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan E-mail address: [email protected] Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812- 8581, Japan E-mail address: [email protected]

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