central projections

Question

Suppose $A$ is a separable $C^*$ algebra(not necessarily unital),and let $I$ be the ideal generated by central projections in $A$,does there exist nonzero pairwise projections in the quotient $A/I$?

Answer 1

No. Take $A=C_0(\mathbb R)$, then the only (central) projection is $0$, so $I=\{0\}$ and $A/I=A$, which has no nonzero projections.

At the other end, take $A=K(H)$, then $I=\{0\}$, and $K(H)/I=K(H)$ has lots and lots of projections.