C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches II

Abstract

Let K be a compact metric space and let ϕ : K → K be continuous. We study a 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 the open set condition and the measure separation condition. This is a generalization of our previous work, in which we studied the case where γ satisfied the finite branch condition.