Minimality and effectiveness of the groupoid associated to a self-similar ultragraph

Abstract

The notion of a self-similar ultragraph $(G,\mathcal{U},\varphi)$ and its $C^*$-algebra $\mathcal{O}_{G,\mathcal{U}}$ were introduced in our recent work, where we proposed inverse semigroup and groupoid models for such $C^*$-algebras as well. In this paper, we investigate minimality and effectiveness of the groupoid of a self-similar ultragraph $(G,\mathcal{U},\varphi)$. In particular, we obtain a result for simplicity of the $C^*$-algebras $\mathcal{O}_{G,\mathcal{U}}$ in a certain case.