On unitary algebras with graded involution of quadratic growth

Let F be a field of characteristic zero. By a *-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of *-graded codimensions {cgri n (A)}n >= 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary *superalgebras having quadratic growth of *-graded codimensions. As a result we obtain that a unitary *-superalgebra with quadratic growth is T2 & lowast;-equivalent to a finite direct sum of minimal unitary *-superalgebras with at most quadratic growth, where at least one *-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums.