Can we construct a non-unital nuclear $C^*$ algebra $A$ such that $I=\bigoplus_n M_n(\Bbb C)$ is an essential proper ideal in $A$?
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Yes. The subalgebra of $\prod M_n$ given by elements $(x_n)_n$ such that $$\lim_n x_{2 n} \in \mathbb{C} 1$$ and $$\lim_n x_{2 n + 1} = 0$$ is nonunital and contains $I$ as an ideal. To see that it is essential just check that if $(x_n)_n$ is annihilated by every finitely supported sequence it has to be $0$. It is a finite extension of $I$, therefore nuclear.
answered Dec 19, 2018 at 12:56
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1$\begingroup$ This is pretty similar to this answer I gave yesterday. $\endgroup$ Commented Dec 19, 2018 at 14:44