Inverse Semigroups Generated by a Free Product of Abelian Semigroups

This article investigates the $$C^{*}$$ -algebras generated by regular representations of free products of abelian semigroups. It provides a comprehensive analysis of the structure of these $$C^{*}$$ -algebras, employing tools from semigroup theory, operator algebras, and harmonic analysis. A criterion for the simplicity of these $$C^{*}$$ -algebras is established, connecting the algebraic properties of the underlying semigroup to the ideal structure of the generated $$C^{*}$$ -algebra. Specifically, the existence of a finite basis for the semigroup is shown to be equivalent to the inclusion of the compact operators as a subalgebra, while the absence of a finite basis is equivalent to the $$C^{*}$$ -algebra being simple. The introduction of monomial indices and the construction of Fell bundles are key techniques used to obtain these results.