Actions of multiplier Hopf algebras

For an action cu of a group G on an algebra R (over C), the crossed product R x(alpha) G is the vector space of R-valued functions with finite support in G, together with the twisted convolution product given by (xi eta)(p) = Sigma(q is an element of G) xi(q)alpha(q)(eta(q(-1) p)) where p is an element of G. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra A on an algebra R, it is possible to make the tensor product R x A into an algebra, by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by R#A. In the group case, the action alpha of G on R yields an action of the group algebra CG as a Hopf algebra on R and the crossed R x(alpha) G coincides with the smash product R#CG. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf algebras to formulate this duality theorem. We prove a result in the case of an algebraic quantum group and its dual. The more general case is only stated and will be proven in a separate paper on coactions. These duality theorems for actions are substantial generalizations of the corresponding theorem for Hopf algebras. Also the techniques that are used here to prove this result are slightly different and simpler.