An Approximation of Semigroup 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 (N₁,N₂,…) consisting of sequences Nᵢ 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.