Minimal algebras with respect to their $*$-exponent

Given an m-tuple (A(1),...,A(m)) of finite dimensional *-simple algebras we introduce a block-triangular matrix algebra with involution, denoted as UT*(A(1),...,A(m)), where each A(i) can be embedded as *algebra. We describe the T*-ideal of R = UT*(A(1),...,A(m)) in terms of the ideals T*(A(i)) and prove that any algebra with involution which is minimal with respect to its *-exponent is *-PI equivalent to R for a suitable choice of (A(1),...,A(m)). Moreover we show that if m = 1 or A(i) = F for all i then R itself is a *-minimal algebra. The assumption for the base field F is characteristic zero.