A generalized Cuntz-Krieger uniqueness theorem for higher rank graphs

TL;DR

This paper proves a new uniqueness theorem for higher rank graph C*-algebras, showing that injectivity of representations can be established without traditional assumptions, by focusing on a distinguished abelian subalgebra.

Contribution

It introduces a generalized Cuntz-Krieger uniqueness theorem that removes the need for aperiodicity or gauge invariance assumptions in higher rank graph C*-algebras.

Findings

Injectivity on a distinguished abelian subalgebra ensures full injectivity.

The proof uses an abstract uniqueness theorem based on faithful states.

The result broadens the class of representations guaranteeing uniqueness.

Abstract

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.