  The main purpose of this paper is to unify the theory of actions of Hopf
algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of
weak multiplier Hopf algebras introduced by A. Van Daele and S. H. Wang. Using
such developed actions, we will define the notion of a module algebra over weak
multiplier Hopf algebras and construct their smash products. The main result is
the duality theorem for actions and their dual actions on the smash product of
weak multiplier Hopf algebras. As an application, we recover the main results
found in the literature for weak Hopf algebras, multiplier Hopf algebras and
groupoids.
