Uniqueness Theorems For Topological Higher-rank Graph C*-algebras

We consider the boundary-path groupoids of topological higher-rank graphs. We show that all such groupoids are topologically amenable. We deduce that the $ C^*$-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph $ C^*$-algebra, and a condition under which it is also purely infinite.