We know that every simple $C^*$-algebra is primitive, say it has a faithful non-zero irreducible representation. The converse is not necessarily true. An counterexample is just the $B(H)$ when $H$ is of infinite dimension.
But if every irreducible representation of $C^*$-algebra $A$ is faithful, whether $A$ is simple?
Thank you for all helps!
Yes. What you do is show that if $A$ is not simple, then there is a non-faithful irrep.
If $J\subset A$ is a non-trivial ideal, then consider an irreducible representation of $A/J$ into $B(H_J)$; then $A\to A/J\to B(H_J)$ is an irreducible representation of $A$ with kernel that at least contains $J$, so it is not faithful.