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.

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. Mathematics Subject Classification (2000): 46L05, 46L55The motivation of this study is our recent research on C * -unique groups in [10]. In fact, one interesting question is when the semi-direct product of a C * -unique group with another group is again C * -unique. This turns out to be related to the following question: Given a C * -dynamical system (A, G, α), under what condition will it be true that any non-zero ideal of A × α G contains a non-zeroα-invariant ideal (α being the dual coaction)? The aim of this paper is to study this question.In fact, in the case of discrete amenable groups acting on compact spaces, Kawamura and Tomiyama gave (in [9]) a complete solution of the above question. In [1], Archbold and Spielberg generalised the main result in [9] to the case of discrete C * -dynamical system. In this article, we are going to present a weaker result but in the case of general locally compact groups. As a corollary, we obtain some equivalent conditions for the primeness of crossed products (in terms of the actions). Moreover, we will also give a brief discussion on the simplicity of crossed products (which is related but does not need the main theorem).

We will consider coactions of discrete groups on C*-algebras and imitate some of the results about compact group actions on C*-algebras. In particular, the crossed product of a reduced coaction e of a discrete amenable group G on A is liminal (respectively, postliminal) if and only if the fixed point algebra of e is. Moreover, we will also consider ergodic coactions on C* -algebras.1991 Mathematics subject classification (Amer. Math. Soc): 46L55.

In this paper, we study coactions and their crossed products by Hopf C*‐algebras defined by multiplicative unitaries. We generalize some results in the case of group actions and coactions. Furthermore, when applied to the group case, some of these generalizations improve the original results. For example, we obtained the following result. Let A be a C*‐algebra with coaction ɛ by S where S is a Hopf C*‐algebra defined by an amenable multiplicative unitary. If A is nuclear or C*‐exact, then so is A Xɛ, rŜ. This implies that if (A, G, α) is any C*‐dynamical system such that A Xα, rG is nuclear or C*‐exact, then so is A.