minimal projection in a $C^*$ algebra

Question

If $(H,\pi)$ is a nonzero representation of a $C^*$ algebra $A$,does there exist a minimal projection $E\in A$ such that $\pi(E)\neq 0$?

Answer 1

No. Let $A=B(H)$ for some Hilbert space $H$, let $\rho:B(H)\to B(H)/K(H)=Q(H)$ be the quotient map, and let $(K,\pi)$ be a non-zero representation of $Q(H)$. Then $(K,\pi\circ\rho)$ is a non-zero representation of $B(H)$ with $\pi\circ\rho(P)=0$ for every finite-rank projection $P$ in $B(H)$, hence for all minimal projections.