Star-Varieties of proper central exponent greater than two

Abstract

Let $F$ be a field of characteristic zero and let ${\mathcal{V}}^{\ast }$ be a variety of associative $F$-algebras with involution *. Associated to ${\mathcal{V}}^{\ast }$ 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 \( c^{*,δ}_n(\mathcal V^*) \). 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 ${\mathcal{V}}^{\ast }.$ In \cite{MR2022} it was proved that $ex{p}^{\ast ,\delta }({\mathcal{V}}^{\ast })={\lim }_{n\to \infty }\sqrt[n]{c_n^{\ast ,\delta }({\mathcal{V}}^{\ast })}$ exists and is an integer called the proper central $\ast$-exponent. The aim of this paper is to study the varieties of associative algebras with involution of proper central $\ast$-exponent greater than two. To this end we construct a finite list of algebras with involution and we prove that if $ex{p}^{\ast ,\delta }({\mathcal{V}}^{\ast })>2$, then at least one of these algebras belongs to ${\mathcal{V}}^{\ast }$.