It's a more recent book, but I've read An introduction to quantum group and duality from Thomas Timmermann. I love how this book is written, although a lot of the proofs are missing. It's more of a year, year-and-a-half course. He closes the book with a treatment of Baaj-skandalis's beautiful result on duality of actions and coactions (which is my favorite topic) and then proceeds to extend results by means of very tough operator algebraic methods to the framework of quantum groupoids.
Works on this area of quantum groupoids is done by Gabriella böhm, ping Xu, Timmermann, Michael Enock, to cite a few. You can arxiv them to find a lot of their work. See also the notion of weak hopf algebras and hopf algebroids.
This "duality theory" they are trying to develop, outside the classical group-vector space framework, is a vast generalization of pontryagin's duality theorem, which states that the bidual of an abelian locally compact group is isomorphic to the said group and a later theory of tannaka-krein duality which evolved in order to remove the commutation hypothesis. These types of theorems ellucidate the representation theory of the algebraic structure, which in turn yields tons of amazing structural results on the algebraic structure itself. So this is a pretty nice thing to study. Remember that function analysis is, roughly speaking, the study of functionals on vector spaces, which are elements of the dual vector space.
Hope I could be of help.