Type $II_\infty $ factors generated by purely infinite simple C*-algebras associated with free groups

Abstract

Let Γ = G1 ∗G2 ∗ ... ∗Gn ∗ ... be a free product of at least two but at most countably many cyclic groups. With each such group Γ we associate a family of C*-algebras, denoted C∗ r (Γ,PΛ) and generated by the reduced group C*-algebra C∗ r Γ and a collection PΛ of projections onto the `2-spaces over certain subsets of Γ. We determine W ∗(Γ,PΛ), the weak closure of C∗ r (Γ,PΛ) in L(`2(Γ)), and use this result to show that many of the C*-algebras in question are non-nuclear.