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

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 $$M_n(\Bbb C)$$.

## 1 Answer

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 $$A$$.)