Let $A$ be a $C^*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?
No. Take $A$ to be $c$, the space of convergent sequences. Its second dual is $l^\infty$, which is atomic, but it is not an ideal of $l^\infty$.