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

Szymański, Wojciech; Zhang, Shuang

doi:10.1090/S0002-9939-99-05074-1

Type $II_\infty$ factors generated by purely infinite simple C*-algebras associated with free groups
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by Wojciech Szymański and Shuang Zhang PDF

Proc. Amer. Math. Soc. 128 (2000), 813-818 Request permission

Abstract:

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.

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