A structure theorem for quasi-Hopf comodule algebras

If $H$ is a quasi-Hopf algebra and $B$ is a right $H$-comodule algebra such that there exists $v:H\rightarrow B$ a morphism of right $H$-comodule algebras, we prove that there exists a left $H$-module algebra $A$ such that $B\simeq A\char93 H$. The main difference when comparing to the Hopf case is that, from the multiplication of $B$, which is associative, we have to obtain the multiplication of $A$, which in general is not; for this we use a canonical projection $E$ arising from the fact that $B$ becomes a quasi-Hopf $H$-bimodule.