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

Abstract

Let \(K\) be a compact metric space and let \(\gamma = (\gamma_1, \dots, \gamma_n)\) be a system of proper contractions on \(K\). We study a C*-algebra \(\mathcal{MC}_{\gamma_1, \dots, \gamma_n}\) generated by all multiplication operators by continuous functions on \(K\) and composition operators \(C_{\gamma_i}\) induced by \(\gamma_i\) for \(i=1, \dots, n\) on a certain \(L^2\) space. Suppose that \(K\) 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 \(\mathcal{MC}_{\gamma_1, \dots, \gamma_n}\) is isomorphic to the Cuntz algebra \(\mathcal{O}_n\) under some conditions.