Cartan subalgebras in self-similar graph $C^{\ast }$-algebras

Abstract

For a self-similar graph $(G,E)$, we find a distinguished subgroupoid of the associated path groupoid ${\mathcal{G}}_{G,E}$ -- the symmetric cycline subgroupoid ${\mathcal{S}}_{\text{sym}}$. If the acting group $G$ is abelian, we show that ${\mathcal{S}}_{\text{sym}}$ is open, abelian, and normal. For $G=\mathbb{Z}$, we describe the dual bundle ${\hat{\mathcal{S}}}_{\text{sym}}$ of ${\mathcal{S}}_{\text{sym}}$ which can be used to provide a different groupoid model for the self-similar graph $C^{\ast }$-algebra ${\mathcal{O}}_{\mathbb{Z},E}\cong C_r^{\ast }({\mathcal{G}}_{\mathbb{Z},E})$. For a large class of self-similar graphs $(\mathbb{Z},E)$, we further prove that ${\mathcal{S}}_{\text{sym}}$ is maximal among open abelian subgroupoids of $\mathrm{Iso}({\mathcal{G}}_{\mathbb{Z},E}{)}^{\circ }$ and closed in ${\mathcal{G}}_{\mathbb{Z},E}$, so that it gives rise to a Cartan subalgebra of ${\mathcal{O}}_{\mathbb{Z},E}$. This result seems new even for genuine actions. Our proofs heavily rely on careful studies of dynamical behaviours of cycline triples of $(\mathbb{Z},E)$ and on a dynamical-flavour classification for the vertices of $E$. Some results hold in more general settings and may be of independent interest.

53 pages