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

Abstract

Let $K$ be a compact metric space and let $φ: K \to K$ be continuous. We study C*-algebra $\mathcal{MC}_φ$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_φ$ induced by $φ$ on a certain $L^2$ space. Let $γ= (γ_1, \dots, γ_n)$ be a system of proper contractions on $K$. Suppose that $γ_1, \dots, γ_n$ 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 $\mathcal{MC}_φ$ is isomorphic to the C*-algebra $\mathcal{O}_γ(K)$ associated with $γ$ under some conditions.