There is a well known fact:$A=\oplus_{i\in I}A_i$($c_0$ direct sum) is an essential ideal $\prod_{i\in I}A_i$($l^\infty$ direct sum),where each $A_i$ is a $C^*$ algebra.
I have two questions:
1.If $\oplus_{i\in I}A_i$ is an essential ideal of a $C^*$ algebra B, can we have B =$\prod_{i\in I}A_i$?
2.If $\oplus_{i\in I}A_i$ is an ideal of B,B has a faithful tracial state,can we conclude that $\oplus_{i\in I}A_i$ is an essential ideal of B?
The answer of the first question is no. Take $A_i = \mathbb C$. Then any compact space in which $\mathbb N$ embeds as a dense open set gives a $C^\ast$-algebra that contains $c_0$ as an essential ideal.
Similarly the second question has also a negative answer. Take $B$ as $\ell^\infty \oplus \mathbb C$. There is a faithful tracial state in $B$ but $c_0$ is not essential in $B$ (if you embed it in the first component)