Relative topological principality and the ideal intersection property for groupoid C*-algebras

We introduce the notion of relative topological principality for a family $\{H_\alpha\}$ of open subgroupoids of a Hausdorff \'etale groupoid $G$. The C*-algebras $C^*_r(H_\alpha)$ of the groupoids $H_\alpha$ embed in $C^*_r(G)$ and we show that if $G$ is topologically principal relative to $\{H_\alpha\}$ then a representation of $C^*_r(G)$ is faithful if and only if its restriction to each of the subalgebras $C^*_r(H_\alpha)$ 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.