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

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 $\mathbb{N}$.