  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.
