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 $A$. 



Universal unital $C^*$ algebra generated by the relations of an n by n projection
This is example (12), pp. 253 in the paper of B. Blackadar, "Shape Theory for C\*-algebras", [Math. Scand. **56** (1985) 249-275](http://www.mscand.dk/article/view/12100/10116), which lays the general theory for universal C\*-algebras. The resulting universal C\*-algebra is called, as you may be expecting, the "noncommutative Grassmannian".


