Varieties of algebras with pseudoinvolution and polynomial growth

Let A be an associative algebra with pseudoinvolution ∗ over an algebraically closed field of characteristic zero and let cn∗(A) be its sequence of ∗-codimensions. We shall prove that such a sequence is polynomially bounded if and only if the variety generated by A does not contain five explicitly described algebras with pseudoinvolution. As a consequence, we shall classify the varieties of algebras with pseudoinvolution of almost polynomial growth, i.e. varieties of exponential growth such that any proper subvariety has polynomial growth and, along the way, we shall give also the classification of their subvarieties. Finally, we shall describe the algebras with pseudoinvolution whose ∗-codimensions are bounded by a linear function.