# Journal of Operator Theory

Volume 75, Issue 2, Spring 2016 pp. 289-298.

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

**Authors**:
Hiroyasu Hamada

**Author institution:** National Institute of Technology, Sasebo College,
Okishin, Sasebo, Nagasaki, 857-1193, Japan

**Summary: **Let $R$ be a rational function of degree at least two,
let $J_R$ be the Julia set of $R$ and let $\mu^\mathrm L$ be the Lyubich measure
of $R$.
We study the $C^*$-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 $L^2(J_R, \mu^\mathrm L)$.
We show that the $C^*$-algebra $\mathcal{MC}_R$ is isomorphic to
the $C^*$-algebra $\mathcal{O}_R (J_R)$ associated with the complex dynamical
system $\{R^{\circ n} \}_{n=1} ^\infty$.

**DOI: **http://dx.doi.org/10.7900/jot.2015mar03.2085

**Keywords: **composition operator, multiplication operator,
Frobenius-Perron operator, $C^*$-algebra, complex dynamical system

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