Polynomial growth and identities of superalgebras and star-algebras

We study associative algebras with 1 endowed with an automorphism or antiautomorphism φ of order 2, i.e., superalgebras and algebras with involution. For any fixed k≥1, we construct associative φ-algebras whose φ-codimension sequence is given asymptotically by a polynomial of degree k whose leading coefficient is the largest or smallest possible.