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

Let $\Gamma = G_{1}*G_{2}*...*G_{n}* ...$ be a free product of at least two but at most countably many cyclic groups. With each such group $\Gamma$ we associate a family of C*-algebras, denoted $C^{*}_{r}(\Gamma,\mathcal{P}_{\Lambda})$ and generated by the reduced group C*-algebra $C^{*}_{r}\Gamma$ and a collection $\mathcal{P}_{\Lambda }$ of projections onto the $\ell ^{2}$-spaces over certain subsets of $\Gamma$. We determine $W^{*}(\Gamma, \mathcal{P}_{\Lambda })$, the weak closure of $C^{*}_{r}(\Gamma, \mathcal{P}_{\Lambda })$ in $\mathcal{L}(\ell ^{2}(\Gamma ))$, and use this result to show that many of the C*-algebras in question are non-nuclear.