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 $\varphi: K \to K$ be continuous. We study C*-algebra $\mathcal{MC}_\varphi$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_\varphi$ induced by $\varphi$ 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 $\varphi$ 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 $\mathcal{MC}_\varphi$ is isomorphic to the C*-algebra $\mathcal{O}_\gamma (K)$ associated with $\gamma$ under some conditions.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2019
- DOI:
- 10.48550/arXiv.1911.10993
- arXiv:
- arXiv:1911.10993
- Bibcode:
- 2019arXiv191110993H
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Functional Analysis
- E-Print:
- 11 pages. arXiv admin note: text overlap with arXiv:1502.02093