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$C^*$-ALGEBRAS OF SELF-SIMILAR ACTION OF GROUPOIDS ON ROW-FINITE DIRECTED GRAPHS
Part of
- Selfadjoint operator algebras
ISNIE YUSNITHA
Journal:

Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023

Published online by Cambridge University Press:

08 November 2022, pp. 150-161

Print publication:

August 2023
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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.