On existence of PI-exponent of algebras with involution

We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of *-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional -PI-exponent. We also construct a family of infinite-dimensional algebras $C_alpha$ such that $exp^ast (C_alpha)$ does not exist