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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.


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

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    $\begingroup$ Just a remark to add to Nik Weaver's answer: since you've asked several questions where you want $I=\bigoplus_n M_n$ to be an essential ideal in some containing algebra $A$, you might want to loook up Busby's theory of extensions, specifically the link to $*$-homomorphisms from a given $B$ into $M(I)/I$. $\endgroup$ – Yemon Choi Dec 20 '18 at 0:28
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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 $\bigoplus M_n$.

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