Toeplitz algebras and {$C\sp *$}-algebras arising from reduced (free) group {$C\sp *$}-algebras

Abstract

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 $l^2(\Omega )$.