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 $\bigoplus M_n$.