We will define two canonical cohomology theories for Hopf C -algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We will then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups will be considered in more details.

In this paper, we will define the reduced cross-sectional C*-algebras of C*-algebraic bundles over locally compact groups and show that if a C*-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional C*-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying C*-algebra is nuclear, then the cross-sectional C*-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups.

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 study amenable unitary corepresentations of Kac algebras (which is defined in analogy to the case of locally compact groups). We also study some sorts of "non-commutative" Reiter's properties. As an application, we find some new equivalent conditions for the amenability of Kac algebras.

We show that the Morita equivalence part of Mansfield's Imprimitivity Theory can be obtained by Green's Imprimitivity Theorem (and duality theory).