My question is originally about the analysis of extensions of $M_n$ by $K(H)$,"$K$-theory for operator algebras"- Blackadar, Exmaple $15.4.1.(b).$
We examine extensions of $M_n$ by $K(H)$, so that we need to look at the Busby invariant, i.e. homomorphism $\tau: M_n\to Q(H)$, where $Q$ denotes the Clakin algebra.
Let $\{\bar{e_{ij}}\}$ be the matrix units in $\tau(M_n)$.
First lift $\{\bar{e_{ii}}\}$ to orthogonal projections $p_{ii}$ in $B(H)$.
Then we get (by some process) $\{e_{ij}\}$ lifted set of matrix units, so that we have $\bar{\tau}:M_n\to B(H)$ s.t. $\pi\circ \bar{\tau}=\tau$, where $\pi$ denotes the canonical quotient map.
$\bar{\tau}$ is not necessarily unital, but $p=\sum e_{ii}$ is a projection of finite co-dimension $k$, i.e. $(1-p)B(H)(1-p)\cong M_k(\Bbb{C})$. This part is not clear to me, why does $p$ must have a finite co-dimension?
Thank you for any help