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

TL;DR

This paper investigates conditions under which the algebra of continuous functions on a compact space forms a Cartan subalgebra within the Kajiwara--Watatani algebra associated with an iterated function system, extending previous results.

Contribution

It provides new sufficient conditions for $C(X)$ to be a Cartan subalgebra or fail to be a maximal abelian subalgebra in the associated algebra.

Findings

$C(X)$ is a Cartan subalgebra under certain conditions.

$C(X)$ may not be a masa if specific criteria are not met.

Extended the class of iterated function systems where these properties hold.

Abstract

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 $\O_\gamma$ associated with $\gamma$. In this paper, we extend their results by providing sufficient conditions under which $C(X)$ becomes a Cartan subalgebra of $\O_\gamma$. Additionally, we present sufficient conditions under which $C(X)$ fails to be a masa of $\O_\gamma$.