On the *-minimality of algebras with involution

In a paper of Di Vincenzo and La Scala (2007) , given a n-tuple (A_1, . . . , A_n) of finite dimensional *-simple algebras over a field of characteristic zero, a block-triangular matrix algebra with involution, denoted by UT_*(A_1,…, A_n), was introduced. More precisely, in that paper it was proved that any finite dimensional algebra with involution which is minimal with respect to its *-exponent is *-PI equivalent to UT_*(A_1,…, A_n) for a suitable choice of the algebras A_i . Motivated by a conjecture stated in the same paper, here we show that UT_*(A_1,…, A_n) is *-minimal when either it is *-symmetric or n = 2.