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

Abstract

Let $X$ be a compact Hausdorff space, and let $γ$ be an iterated function system on $X$. Kajiwara and Watatani showed that if $γ$ 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 $Ø_γ$ associated with $γ$. In this paper, we extend their results by providing sufficient conditions under which $C(X)$ becomes a Cartan subalgebra of $Ø_γ$. Additionally, we present sufficient conditions under which $C(X)$ fails to be a masa of $Ø_γ$.