Trace identities and almost polynomial growth

In this paper we study algebras with trace and their polynomial identities over a field of characteristic 0. We consider two commutative matrix algebras: $D_2$, the algebra $2\times 2$ diagonal matrices $C_2$, $2 \times generated by $e_{11}+e_{22}$ $e_{12}$. describe all possible traces on these corresponding codimensions. Moreover characterize varieties growth finite dimensional algebra. As consequence, see that variety is either or exponential.