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

A class of C*-algebraic locally compact quantum groupoids Part II: Main theory