Varieties of Algebras with Superinvolution of Almost Polynomial Growth

Let A be an associative algebra with superinvolution ? over a field of characteristic zero and let (c_{n}^{ast }(A)) be its sequence of corresponding ?-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ?-codimensions between polynomial and exponential is allowed.