Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs

Abstract

We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs, and show that the $C^{\ast }$-algebras defined are coherent with the existing theory.

arXiv comment13 pages

References in corpus (5)

• Higher rank graph C*-algebras
• Product systems of graphs and the Toeplitz algebras of higher-rank graphs
• On the K-theory of higher rank graph C*-algebras
• Graph C*-algebras and Z/2Z-quotients of quantum spheres
• A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras II, examples

Cited by in corpus (8)

• Cartan subalgebras and the UCT problem, II
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs
• Simplicity of twisted C*-algebras of Deaconu--Renault groupoids
• Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras
• Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras
• Approaching the UCT problem via crossed products of the Razak-Jacelon algebra
• Coverings of skew-products and crossed products by coactions
• Sheaves of Measures and KMS-Weights on Topological Graph Algebras