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 $\phi :K\to K$ be continuous. We study C*-algebra ${\mathcal{MC}}_{\phi }$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_{\phi }$ induced by $\phi$ on a certain $L^2$ space. Let $\gamma =({\gamma }_1,\dots ,{\gamma }_n)$ be a system of proper contractions on $K$. Suppose that ${\gamma }_1,\dots ,{\gamma }_n$ are inverse branches of $\phi$ and $K$ is self-similar. We consider the Hutchinson measure ${\mu }^H$ of $\gamma$ and the $L^2$ space $L^2(K,{\mu }^H)$. Then we show that the C*-algebra ${\mathcal{MC}}_{\phi }$ is isomorphic to the C*-algebra ${\mathcal{O}}_{\gamma }(K)$ associated with $\gamma$ under some conditions.

11 pages. arXiv admin note: text overlap with arXiv:1502.02093