C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches

Abstract

Let [Formula: see text] be a compact metric space and let [Formula: see text] be continuous. We study a [Formula: see text]-algebra [Formula: see text] generated by all multiplication operators by continuous functions on [Formula: see text] and a composition operator [Formula: see text] induced by [Formula: see text] on a certain [Formula: see text] space. Let [Formula: see text] be a system of proper contractions on [Formula: see text]. Suppose that [Formula: see text] are inverse branches of [Formula: see text] and [Formula: see text] is self-similar. We consider the Hutchinson measure [Formula: see text] of [Formula: see text] and the [Formula: see text] space [Formula: see text]. Then we show that the [Formula: see text]-algebra [Formula: see text] is isomorphic to the [Formula: see text]-algebra [Formula: see text] associated with [Formula: see text] under some conditions.