Relative Topological Principality and the Ideal Intersection Property for Groupoid C*-Algebras

We introduce the notion of relative topological principality for a family \(\{H_\alpha \}\) $\{H_{\alpha }\}$ of open subgroupoids of a Hausdorff étale groupoid G. The C*-algebras \(C^*_r(H_\alpha )\) $C_r^{\ast }(H_{\alpha })$ of the groupoids \(H_\alpha \) $H_{\alpha }$ embed in \( C^*_r(G)\) $C_r^{\ast }(G)$ and we show that if G is topologically principal relative to \(\{H_\alpha \}\) $\{H_{\alpha }\}$ then a representation of \(C^*_r(G)\) $C_r^{\ast }(G)$ is faithful if and only if its restriction to each of the subalgebras \(C^*_r(H_\alpha )\) $C_r^{\ast }(H_{\alpha })$ is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic.