Duality for weak multiplier Hopf algebras with sufficiently many integrals

We study duality of regular weak multiplier Hopf algebras with sufficiently many integrals. This generalizes the well-known duality of algebraic quantum groups. We need to modify the definition of an integral in this case. It is no longer true that an integral is automatically faithful and unique. Therefore we have to work with a faithful set of integrals. We apply the theory to three cases and give some examples. First we have the two weak multiplier Hopf algebras associated with an infinite groupoid (a small category). Related we answer a question posed by Nicolás Andruskiewitsch about double groupoids. Finally, we also discuss the weak multiplier Hopf algebras associated to a separability idempotent.