-ALGEBRAS OF SELF-SIMILAR ACTION OF GROUPOIDS ON ROW-FINITE DIRECTED GRAPHS

Source abstract

Abstract For amenable discrete groupoids $\mathcal {G}$ and row-finite directed graphs E , let $(\mathcal {G},E)$ be a self-similar groupoid and let $C^*(\mathcal {G}, E)$ be the associated $C^*$ -algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$ . Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.