C*-algebras generated by multiplication operators and composition operators with rational symbol

Abstract

Let $R$ be a rational function of degree at least two, let $J_R$ be the Julia set of $R$ and let ${\mu }^L$ be the Lyubich measure of $R$. We study the C${}^{\ast }$-algebra ${\mathcal{MC}}_R$ generated by all multiplication operators by continuous functions in $C(J_R)$ and the composition operator $C_R$ induced by $R$ on $L^2(J_R,{\mu }^L)$. We show that the C${}^{\ast }$-algebra ${\mathcal{MC}}_R$ is isomorphic to the C${}^{\ast }$-algebra ${\mathcal{O}}_R(J_R)$ associated with the complex dynamical system $\{R^{\circ n}{\}}_{n=1}^{\infty }$.

9 pages

References in corpus (1)

• A Semigroup Composition C*-algebra