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) $.