Toeplitz algebras and \(C^{\ast}\)-algebras arising from reduced (free) group \(C^{\ast}\)-algebras (Q1425776)

Let \(\Gamma\) be a free group of \(n\) generators and \(r<n<+\infty\). Given an infinite subset \(\Omega\) of \(\Gamma\) such that \(\Gamma \setminus \Omega\) is also infinite, denote by \(P\) the projection of the Hilbert space \(l^2(\Gamma)\) onto the subspace \(l^2(\Omega)\). The author proves 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 nontrivial stable closed ideals of real rank zero. He also gives a detailed analysis of the Toeplitz algebra generated by the restrictions of operators in \(C_r^*(\Gamma, P)\) on the subspace \(l^2(\Omega)\).