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

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

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