Does there exist a non-unital nuclear $C^*$ algebra $A$ of $\prod_nM_n(\Bbb C)$ such that $A$ properly contains $\oplus_n M_n(\Bbb C)$ and each element $(x_n)\not \in \oplus_n M_n(\Bbb C)$ we have $\lim_ntr_n(x_n)=0$,where $tr$ is the unique tracial state on $M_n(\Bbb C)$.

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Sure, let $A = (\bigoplus M_n ) + \mathbb{C}\cdot P$ where $P$ is a projection of the form $P = (p_n)$ with each $p_n \in M_n$ a rank 1 projection. This is a one-dimensional extension of $\bigoplus M_n$, and so it is nuclear. (If $I$ and $A/I$ are nuclear then so is $A$.)