In this article, we will give the definition of the Ext-functor for the category MS of 13 (completely bounded) counital right S-comodules (where S is a counital Hopf C*-algebra that is either unital or nuclear) using the derived functor approach. Moreover, we will give a brief discussion on the injective objects in M15 S.

We will study some permanence properties of C*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C*-unique group. Moreover, any amenable discrete group is a retract of a discrete C*-unique group.

In this paper, we study the duality theory of Hopf C -algebras in a general "Hilbert-space-free" framework. Our particular interests are the "full duality" and the "reduced duality". In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf C*-algebra. This study of reduced duality and Fourier algebra is found to be useful in the study of other aspects of Hopf C -algebras (see e.g. [12], [13] and [14]).

By giving an interesting characterisation of amenable multiplicative unitaries, we show, in a very simple way, that bicrossproducts of amenable locally compact groups are both amenable and coamenable.

In this paper, we will give a natural definition for morphisms between multiplicative unitaries. We will then discuss some equivalences of this definition and some interesting properties of them. Moreover, we will define normal sub-multiplicative unitaries for multiplicative unitaries of discrete type and prove an imprimitivity type theorem for discrete multiplicative unitaries.