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.

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 will prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on C*-algebras.

Suppose that (A, G, α) is a C -dynamical system such that G is of polynomial growth. If A is finite dimensional, we show that any element in K (G; A) has slow growth and that L1 (G,A) is *-regular. Furthermore, if G is discrete and π is a "nice representation" of A, we define a new Banach-algebra l1 (G,A) which coincides with l1 (G; A) when A is finite dimensional. We also show that any element in K (G; A) has slow growth and l1 (G,A) is-regular.

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]).