Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

Summary Let Γ=〈g ₁〉*〈g ₂〉*...*〈g _n 〉*... be a free product of cyclic groups with generators {g _i }, andC _r ^* (Γ,℘ _Λ) be the C*-algebra generated by the reduced group C*-algebraC _r ^* Γ and a set of projectionsP _gL associated with a subset Λ of {g _i }. We prove the following: (1)C _r ^* (Γ,℘ _Λ) is *-isomorphic to the reduced cross product for certain Hausdorff compact spaceX _Λ constructed from Γ and its boundary ∂Γ. (2)C _r ^* (Γ,℘ _Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the pair (Γ, Λ). (3)C _r ^* (Г,℘ _Λ) is nuclear if and only if the subgroup Γ_Λ generated by {g _i }/Λ is amenable. Partially supported by RMC grant 45/290/603 from the University of Newcastle Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati