Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$?
Well, there are two projections equivalent to $1$ such that $pq=0$ but what can we can we say about $p+q$?
You cannot always get p + q = 1. We can see this using K-theory.
Suppose p,q are orthogonal projections, both Murray von-Neumann equivalent to 1, such that p + q = 1. Let [-] denote the class of a projection in K_0. Then we have
[1] = [p+q] = [p] + [q] = [1] + [1]
Thus [1] = 0. But the class of the unit is not always zero in K_0 of a purely infinite C*-algebra. For instance it is not zero in the Cuntz algebra O_3.