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 C R on the Hardy space and the C * -algebra O R (J R ) associated with the complex dynamical system (R •n ) n on the Julia set J R . We study the C * -algebra T C R generated by both the composition operator C R and the Toeplitz operator T z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C * -algebra O R (J R ), which is simple and purely infinite.

Let R be a finite Blaschke product. We study the C * -algebra T C R generated by both the composition operator C R and the Toeplitz operator Tz on the Hardy space. We show that the simplicity of the quotient algebra OC R by the ideal of the compact operators can be characterized by the dynamics near the Denjoy-Wolff point of R if the degree of R is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of OC R such that R is a finite Blaschke product of degree at least two and the Julia set of R is the unit circle, using the Kirchberg-Phillips classification theorem.

Let K be a compact metric space and let ϕ : K → K be continuous. We study C * -algebra MCϕ generated by all multiplication operators by continuous functions on K and a composition operator Cϕ induced by ϕ on a certain L 2 space. Let γ = (γ 1 , . . . , γn) be a system of proper contractions on K. Suppose that γ 1 , . . . , γn are inverse branches of ϕ and K is self-similar. We consider the Hutchinson measure µ H of γ and the L 2 space L 2 (K, µ H ). Then we show that the C * -algebra MCϕ is isomorphic to the C * -algebra Oγ(K) associated with γ under some conditions. 2010 Mathematics Subject Classification. Primary 46L55, 47B33; Secondary 28A80, 46L08.

No abstract

Let [Formula: see text] be a compact metric space and let [Formula: see text] be continuous. We study a [Formula: see text]-algebra [Formula: see text] generated by all multiplication operators by continuous functions on [Formula: see text] and a composition operator [Formula: see text] induced by [Formula: see text] on a certain [Formula: see text] space. Let [Formula: see text] be a system of proper contractions on [Formula: see text]. Suppose that [Formula: see text] are inverse branches of [Formula: see text] and [Formula: see text] is self-similar. We consider the Hutchinson measure [Formula: see text] of [Formula: see text] and the [Formula: see text] space [Formula: see text]. Then we show that the C[Formula: see text]-algebra [Formula: see text] is isomorphic to the C[Formula: see text]-algebra [Formula: see text] associated with [Formula: see text] under some conditions.