# construct a non-unital nuclear $C^*$ algebra

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]

• 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 $M(I)/I$. – Yemon Choi Dec 20 '18 at 0:28

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 $$\bigoplus M_n$$.