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 $M\_n(\Bbb C)$.



a nuclear $C^*$-subalgebra in $\prod_n M_n(\Bbb C)$
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 $A$.)


