A generalized Cuntz-Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k-graph C⁎-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C⁎-algebra, it is sufficient that the representation be injective on a distinguished abelian C⁎-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C⁎-algebra, each of which is the unique extension of a state on the distinguished abelian C⁎-subalgebra.