An Approximation of Semigroup $$\boldsymbol{C^{*}}$$-Algebras

We consider the semigroup $$C^{*}$$ -algebras generated by the left regular representations of the free products for countable families of semigroups of rational numbers. These free products are constructed from collections $$(N_{1},N_{2},\ldots)$$ consisting of sequences $$N_{i}$$ of arbitrary positive integers. It is proved that the semigroup $$C^{*}$$ -algebras are the inductive limits for sequences consisting of the copies for the Cuntz algebra $$\mathcal{O}_{\infty}$$ and its $$\ast$$ -endomorphisms constructed from the collections $$(N_{1},N_{2},\ldots)$$ . Using this result, we establish that the semigroup $$C^{*}$$ -algebras for the free products are simple. Moreover, it is also shown that the universal $$C^{*}$$ -algebra $$\mathcal{O}_{\infty}$$ is isomorphic to the semigroup $$C^{*}$$ -algebra generated by the representation of the free product for the countable family of copies for the additive semigroup of all natural numbers.