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 $\ast$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $\ast$-PI-exponent. We also construct a family of infinite-dimensional algebras $C_{\alpha }$ such that ${\mathrm{e}\mathrm{x}\mathrm{p}}^{\ast }(C_{\alpha })$ does not exist.