Assume that $\Gamma$ is a free group on $n$ generators, where
$2\le n< +\infty$.
Let $\Omega $ be an infinite subset of $\Gamma$ such that $\Gamma
\setminus \Omega$ is also infinite, and let $P$ be the
projection on the subspace $l^2(\Omega )$ of $l^2(\Gamma )$.
We prove that, for some choices of $\Omega$,
the C*-algebra $C^*_r(\Gamma ,P)$
generated by the reduced group C*-algebra $C^*_r\Gamma$ and
the projection $P$ has exactly two non-trivial, stable, closed
ideals of real rank zero.
We also give a detailed analysis of
the Toeplitz algebra generated by the restrictions of operators
in $C^*_r(\Gamma ,P)$ on the subspace $l^2(\Omega )$.