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

Is there a non-unitary nuclear? $$C ^ *$$ algebra $$A$$ of $$prod_nM_n ( Bbb C)$$ such that $$A$$ contains properly $$oplus_n M_n ( Bbb C)$$ and each element $$(x_n) not in oplus_n M_n ( Bbb C)$$ we have $$lim_ntr_n (x_n) = 0$$,where $$tr$$ it is the only tracial state in $$M_n ( Bbb C)$$.