Nonamenable groups whose reduced group C*-algebras are not pure

RESUMO

We exhibit nonamenable groups whose reduced group C*-algebras are not pure. More precisely, if $\Gamma$ is any countably infinite discrete group, then the reduced group C*-algebra of the restricted wreath product $(\mathbb{Z}/2\mathbb{Z}) \wr \Gamma$ has an ideal-quotient isomorphic to $\mathcal{K}(\ell^2(\Gamma))$. It therefore fails to be nowhere scattered and, in particular, is not pure. Taking $\Gamma$ to be any nonamenable group, we thereby obtain a negative answer to a question of Thiel.