Cartan subalgebras of C*-algebras associated with iterated function systems

Let $X$ be a compact Hausdorff space, and let $\gamma$ be an iterated function system on $X$. Kajiwara and Watatani showed that if $\gamma$ is self-similar and satisfies the open set condition and some additional technical conditions, $C(X)$ is a maximal abelian subalgebra of the Kajiwara–Watatani algebra ${\text{Ø}}_{\gamma }$ associated with $\gamma$. In this paper, we extend their results by providing sufficient conditions under which $C(X)$ becomes a Cartan subalgebra of ${\text{Ø}}_{\gamma }$. Additionally, we present sufficient conditions under which $C(X)$ fails to be a masa of ${\text{Ø}}_{\gamma }$.