Essential ideals in sums of matrix algebras

Question

Given the $C^*$-algebra $A=\prod_{n}M_n(\Bbb C)$, how many essential ideals has $A$? Is there a unique one ?

Answer 1

If the product is finite, there are no essential ideals, as the only way to obtain an ideal is to zero a coordinate.

If the product is infinite, there are infinitely many essential ideals. Explicitly, given any free ultrafilter $\omega\in\beta\mathbb N\setminus\mathbb N$, let $$ J_\omega=\{x\in A:\ \lim_{n\to\omega}\operatorname{tr}_n(x_n^*x_n)=0\}. $$ Any ideal contains $0\oplus\cdots\oplus M_n(\mathbb C)\oplus 0\oplus\cdots$, so it intersects $J_\omega$; so $J_\omega$ is essential.