The Drinfelʹd double of multiplier Hopf algebras

Let A,B be a pairing of two regular multiplier Hopf algebras A and B. The Drinfelʹd double associated to this pairing is constructed by using appropriate representations of A and B on the same vector space B A. We realize the Drinfelʹd double, denoted by D, as an algebra of operators on the vector space B A. In the case that A,B is a multiplier Hopf *-algebra pairing, we prove that D is again a multiplier Hopf *-algebra. If A and B carry positive integrals, we prove that D also has a positive integral. This proof is not given before.