Crossed products for weak Hopf algebras with coalgebra splitting

In this paper we prove that if is a morphism of weak Hopf algebras which is split as a coalgebra morphism, then there exists a subalgebra BH of B, morphisms , and an isomorphism of algebras and right H-comodules , being BH×H a subobject of BH H with its algebra structure defined by a crossed product. Also, we obtain the dual results and as a consequence we prove the Radfordʹs theorem for weak Hopf algebras.