On the ∗-polynomial identities of M1,1(E)

In this paper we consider the algebra M1,1(E) endowed with the involution ∗ induced by the transposition superinvolution of the superalgebra M1,1(F) of 2×2-matrices over the field F. We study the ∗-polynomial identities for this algebra in the case of characteristic zero. We describe a finite set generating the ideal of its ∗-identities. We also consider Mn(E), the algebra of n×n matrices over the Grassmann algebra E. We prove that for a large class of involutions defined on it any ∗-polynomial identity is indeed a polynomial identity. A similar result holds for the verbally prime algebra Mk,l(E).