On the *-polynomial identities of a class of *-minimal algebras

The problem of classifying the finite dimensional *-minimal algebras up to *-PI equivalence has been recently faced by Di Vincenzo and Spinelli. Essentially, if A is a finite dimensional *-minimal algebra over the field F then there exists an n-tuple (A_1,…, A_n) of *-simple algebras allowing the construction of a block-matrix algebra UT_*(A_1,…,A_n) which is *-PI equivalent to A, that is the algebras satisfy the same *-polynomial identities. The simplest case is when A_i = F, for all i. In this case we denote by U_n the algebra UT_*(F,…, F) a sub-algebra of the full matrix algebra M_2n(F). In the present article, we study the *-polynomial identities of U_n. We prove that T_*(U_n) is generated by a single explicit polynomial as soon as F is an infinite field of characteristic different from 2. Moreover, in the case char. F = 0, we describe the structure of T_*(U_n) under the action of general linear groups