Take generators $a_{ij}$ for $1\le i,j\le n$. Impose relations which come from the matrix \begin{eqnarray*} P=\left(\begin{array}{ccc}a_{11} & \dots & a_{1n} \\\vdots & \ddots & \vdots \\a_{n1} & \dots & a_{nn}\end{array}\right) \end{eqnarray*} being a Hermitian projection, i.e. $a_{ij}^*=a_{ji}$ and $\sum_j a_{ij}\,a_{jk}=a_{ik}$. Is there a unital universal $C^*$ algebra with these generators and relations? If not, what can be said?

This question comes from looking at `spaces' associated to finitely generated projective modules in noncommutative geometry.

Take generators $a_{ij}$ for $1\le i,j\le n$. Impose relations which come from the matrix \begin{eqnarray*} P=\left(\begin{array}{ccc}a_{11} & \dots & a_{1n} \\\vdots & \ddots & \vdots \\a_{n1} & \dots & a_{nn}\end{array}\right) \end{eqnarray*} being a Hermitian projection, i.e. $a_{ij}^*=a_{ji}$ and $\sum_j a_{ij}\,a_{jk}=a_{ik}$. Is there a unital universal $C^*$ algebra with these generators and relations? If not, what can be said?

Why is this algebra interesting? The answer is that it might (in some sense) classify finitely generated projective modules over a local $C^*$ algebra $A$. Such modules give projections in $M_n(A)$, and an argument from Blackadar's book shows that these can be made Hermitian. But such a Hermitian projection is just a map from the algebra with generators and relations defined above to $A$.