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

Abstract: 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.