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 (N₁, N₂, ) consisting of sequences N₈ 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 O_ and its -endomorphisms constructed from the collections (N₁, N₂, ). 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 O_ 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.
Gumerov et al. (Sun,) studied this question.