# Universal unital $C^*$ algebra generated by the relations of an n by n projection

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 $A$.

## 2 Answers

This is example (12), pp. 253 in the paper of B. Blackadar, "Shape Theory for C*-algebras", Math. Scand. 56 (1985) 249-275, which lays the general theory for universal C*-algebras. The resulting universal C*-algebra is called, as you may be expecting, the "noncommutative Grassmannian".

• Thanks, that is just what I was looking for. Blackadar in turn refers to a paper by L. Brown "Ext of certain free product C* algebras", J. Operator Theory 6. I have some reading to do. – Edwin Beggs Jun 4 '14 at 22:27

This does not answer directly the question but may be of related interest...

If you take a $2\times 2$ projector with the additional condition that its $0^{th}$ Hochschild Chern class is trivial $ch_0(P)=0$ then what you get is that $P$ is forced to be of the form $$\left(\begin{array}{cc}t&z\\ z^*&1-t\end{array}\right)$$ with relations: $[t,z]=0$, $[z,z^*]=0$, $z^*z=t-t^2$ wich is basically the classical $2$--sphere (see http://arxiv.org/abs/math/0011193, page 29). Basing on this remark in a series of papers by Connes, Connes and Landi, Connes and Dubois-Violette constructed examples of nc geometries from matrix elements of projectors to which vanishing conditions for Chern classe were added.

• That is interesting - also in the case where the diagonals sum to $1+\lambda$ you get the noncommutative fuzzy sphere with parameter $\lambda$. I am not sure who first pointed that out. I shall do a bit more of an edit above to explain why these algebras are so interesting in the general case. – Edwin Beggs Jun 4 '14 at 16:17