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 will prove that if A is a C*-algebra with an effective coaction by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

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

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.

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]).

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 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.

In this article, we will give a chara+M40cterization of Banach bimodules over C*-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule regular).

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.