Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras

Abstract

We investigate $C^{\ast }$-algebras associated with row-finite topological higher-rank graphs with no source, which are based on product system $C^{\ast }$-algebras. We prove the Cuntz-Krieger uniqueness theorem, and provide the condition of simplicity and purely infiniteness of our algebras. We give examples of non-discrete topological higher-rank graphs whose $C^{\ast }$-algebras contain Cuntz's $ax+b$-semigroup $C^{\ast }$-algebra over $\mathbb{N}$.

18 pages, 1 figure, Version 2 Comments: Minor changes and a few small typos corrected

References in corpus (4)

• C*-algebras associated to product systems of Hilbert bimodules
• The Regular C*-algebra of an Integral Domain
• Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs
• Iterating the Pimsner construction