Star-Varieties of proper central exponent greater than two

Let $F$ be a field of characteristic zero and let $\mathcal V^*$ be a variety of associative $F$-algebras with involution *. Associated to $\mathcal V^*$ are three sequences: the sequence of $*$-codimensions $c^{*}_n(\mathcal V^*)$, the sequence of central $*$-codimensions $c^{*,z}_n(\mathcal V^*)$ and the sequence of proper central $*$-codimensions $c^{*,\delta}_n(\mathcal V^*)$. These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomial and the proper central *-polynomial of any generating algebra with involution $A$ of $\mathcal V^*.$ In \cite{MR2022} it was proved that $exp^{*,\delta}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,\delta}(\mathcal V^*)}$ 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 $exp^{*,\delta}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$.