$G$-identities on associative algebras

Let $R$ be an algebra over a field and $G$ a finite group of automorphisms and anti-automorphisms of $R$. We prove that if $R$ satisfies an essential $G$-polynomial identity of degree $d$, then the $G$-codimensions of $R$ are exponentially bounded and $R$ satisfies a polynomial identity whose degree is bounded by an explicit function of $d$. As a consequence we show that if $R$ is an algebra with involution $*$ satisfying a $*$-polynomial identity of degree $d$, then the $*$-codimensions of $R$ are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case $R$ must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.