Star-varieties of proper central exponent greater than two

Let F be a field of characteristic zero and let ⁎ V ⁎ be a variety of associative F -algebras with involution *. Associated with ⁎ V ⁎ are three sequences: the sequence of ⁎-codimensions ⁎ ⁎ c n ⁎ ( V ⁎ ) , the sequence of central ⁎-codimensions ⁎ ⁎ c n ⁎ , z ( V ⁎ ) and the sequence of proper central ⁎-codimensions ⁎ ⁎ c n ⁎ , δ ( V ⁎ ) . These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomials and the proper central *-polynomials of any generating algebra with involution A of ⁎ V ⁎ . In [22] it was proved that ⁎ ⁎ ⁎ ⁎ e x p ⁎ , δ ( V ⁎ ) = lim n → ∞ ⁡ c n ⁎ , δ ( V ⁎ ) n exists and is an integer called the proper central ⁎-exponent. The aim of this paper is to study the varieties of associative algebras with involution of proper central ⁎-exponent greater than two. To this end we construct a finite list of algebras with involution and we prove that if ⁎ ⁎ e x p ⁎ , δ ( V ⁎ ) > 2 , then at least one of these algebras belongs to ⁎ V ⁎ .