Let $I=\bigoplus\_n M\_n(\Bbb C)$,can we construct a non-unital $C^\*$ algebra $A$ such that $I$ is essential in $A$ and $A/I\cong K(H)$ for some separable infinite Hilbert space.




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[note added by YC: this question is a minor variant on previous questions by the OP, see [construct a non-unital nuclear $C^\*$ algebra](https://mathoverflow.net/questions/319081/) and Nik Weaver's comment]



construct a non-unital nuclear $C^*$ algebra
For each $k$, each $x \in M\_k$, and each $n \geq k$, let $x^{(n)} \in M\_n$ be the matrix whose upper left $k \times k$ corner is $x$ and which is zero elsewhere. For $n < k$ let $x^{(n)} = 0$.


Let $A$ be the norm closure of the \*-algebra $A\_0$ generated by $\bigoplus M\_n$ and, for all $k \in \mathbb{N}$ and $x \in M\_k$, the sequence $(x^{(n)}) \in \prod M\_n$. Every element $(x\_n)$ of $A\_0$ has the property that there exists $k \in \mathbb{N}$ and $y \in M\_k$ such that $\|x\_n - y^{(n)}\| \to 0$. So there is a nonexpansive $\*$-homomorphism from $A\_0$ to the finite rank operators on $l^2$ which sends $(x\_n)$ to the operator which is $y$ in the upper left $k\times k$ corner and zero elsewhere. Since this map is nonexpansive, it extends to a $\*$-homomorphism from $A$ onto $K(l^2)$, whose kernel is $\bigoplus M\_n$.


