Uniqueness theorems for topological higher-rank graph C*-algebras

Abstract

We consider the boundary-path groupoids of topological higher-rank graphs. We show that the 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.

Comments for version 3: proofs of uniqueness theorems substantially streamlined and errors in the proof of amenability corrected. New results on simplicity and pure infinite added. 15 pages

References in corpus (4)

• Graph inverse semigroups, groupoids and their C*-algebras
• C*-algebras associated to product systems of Hilbert bimodules
• Non-Hausdorff groupoids
• Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras

Cited by in corpus (5)

• Cartan subalgebras and the UCT problem, II
• Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras
• Aperiodicity Conditions in Topological $k$-Graphs
• Twisted Topological Graph Algebras Are Twisted Groupoid C*-algebras
• Purely Infinite Totally Disconnected Topological Graph Algebras