Motivated by the purely algebraic notion of weak multiplier Hopf algebras, Van Daele (Leuven) and the author developed a C*-algebraic framework for a subclass of quantum groupoids (``quantum groupoids of separable type''). From the axioms, it is possible to construct certain multiplicative partial isometries and the antipode map.
In this talk, going the other way, we will explore possible conditions to be given on a partial isometry $W$ such that $W$ encodes information about a dual pair of C*-algebraic quantum groupoids. The conditions include the pentagon equation, but more is needed. As a special case, when $W$ is a unitary, it will become a multiplicative unitary in the sense of Baaj, Skandalis. Finally, I will propose definitions that generalize the ``regularity'' and the ``manageability'' of a multiplicative unitary.