A Characterization of Superalgebras with Pseudoinvolution of Exponent 2

Let A be a superalgebra endowed with a pseudoinvolution ∗ over an algebraically closed field of characteristic zero. If A satisfies an ordinary non-trivial identity, then its graded ∗-codimension sequence cn∗(A), n = 1,2,…, is exponentially bounded (Ioppolo and Martino (Linear Multilinear Algebra 66(11), 2286–2304 2018). In this paper we capture this exponential growth giving a positive answer to the Amitsur’s conjecture for this kind of algebras. More precisely, we shall see that the limn→∞cn∗(A)n exists and it is an integer, denoted exp ∗(A) and called graded ∗-exponent of A. Moreover, we shall characterize superalgebras with pseudoinvolution according to their graded ∗-exponent.