\(C^*\)-algebra generated by four projections with sum equal to 2

Abstract

We describe the \(C^*\)-algebra generated by four orthogonal projections \(p_1, p_2, p_3, p_4\), satisfying the linear relation \(p_1+p_2+p_3+p_4=2I\). The simplest realization by \(2\times 2\)-matrix-functions over the sphere \(S^2\) is given.