  This is Part II in our multi-part series of papers developing the theory of a
subclass of locally compact quantum groupoids ("quantum groupoids of separable
type"), based on the purely algebraic notion of weak multiplier Hopf algebras.
The definition was given in Part I. The existence of a certain canonical
idempotent element $E$ plays a central role. In this Part II, we develop the
main theory, discussing the structure of our quantum groupoids. We will
construct from the defining axioms the right/left regular representations and
the antipode map.
