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
Contents
Full-Text PDF