Self-similar group actions on ultragraphs and associated $C^*$-algebras
Hossein Larki, Najmeh Rajabzadeh-Hasiri
TL;DR
This work extends the Exel–Pardo self-similar graph framework to ultragraphs by introducing self-similar group actions on ultragraphs and constructing the universal $C^*$-algebras ${\mathcal{O}}_{G,\mathcal{U}}$ via a $(G,\mathcal{U})$-family. It develops an inverse semigroup ${\mathcal{S}}_{G,\mathcal{U}}$ and its tight groupoid ${\mathcal{G}}_{\text{tight}}({\mathcal{S}}_{G,\mathcal{U}})$ to model ${\mathcal{O}}_{G,\mathcal{U}}$ as a tight groupoid $C^*$-algebra, showing ${\mathcal{O}}_{G,\mathcal{U}} \cong C^*_{\text{tight}}({\mathcal{S}}_{G,\mathcal{U}}) \cong C^*(\mathcal{G}_{\text{tight}}({\mathcal{S}}_{G,\mathcal{U}}))$. The paper provides a detailed description of the tight spectrum, the action on it, and the resulting groupoid; it also analyzes when the groupoid is Hausdorff and when the inverse semigroup is $E^*$-unitary or strongly $E^*$-unitary, establishing equivalences with pseudo freeness. These results extend Exel–Pardo's framework and connect ultragraph algebras to Exel–Laca algebras via new constructions and examples, highlighting the broader landscape of self-similar operator algebras.
Abstract
As a generalization of the Exel-Pardo's notion of self-similar graph, we introduce self-similar group actions on ultragraphs and their $C^*$-algebras. We then approach to the $C^*$-algebras by inverse semigroup and tight groupoid models.