Boundaries of Reduced Free Group C*-Algebras
  We prove that the crossed product $C^*$-algebra $C^*_r(\Gamma,\partial\Gamma)$ of a free group $\Gamma$ with its boundary $\partial\Gamma$ naturally sits between the reduced group $C^*$-algebra $C^*_r\Gamma$ and its injective envelope $I(C^*_r\Gamma)$. In other words, we have natural inclusion $C^*_r\Gamma \subset C^*_r(\Gamma,\partial\Gamma) \subset I(C^*_r\Gamma)$ of $C^*$-algebras. 
Narutaka Ozawa