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.

7 pages. arXiv admin note: substantial text overlap with arXiv:2109.08835

References in corpus (3)

• Toeplitz-Composition C*-Algebras
• C^*-algebras associated with self-similar sets
• A Semigroup Composition C*-algebra