Invariant ideals of twisted crossed products

We will prove a result concerning the inclusion of non-trivial invariant ideals inside nontrivial 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. The motivation of this study is our recent research on C-unique groups (or Fourier 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×αG contains a non-zero α̂-invariant ideal (α̂ 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). Throughout this article, A is a C-algebra, G is a locally compact group and (β, u) is a twisted action of G on A (in the sense of Busby and Smith; see [16, 2.1]). Remark 1 Suppose that A is separable and G is second countable. (a) By the stabilisation trick of Packer and Raeburn ([16, 3.4]), there is a canonical continuous action of G on Prim(A) ∼= Prim(A ⊗ K(L(G))) which is the same as the canonical one defined by β. This means that the canonical action of G on Prim(A) is continuous. For any I ∈ Prim(A), the quasi-orbit, Q(I), of I is defined to be the set {I ′ ∈ Prim(A) : G · I = G · I ′}. Moreover, (β, u) is said to be free if Prim(A) = ∅ for any t 6= e. (b) We recall the following materials from [17, §2]. An ideal I of A is said to be β-invariant if βt(I) ⊆ I for any t ∈ G. Note that I is β-invariant if and only if I ⊗ K(L(G)) is invariant under the (ordinary) action given by [16, 3.4]. If I is β-invariant, we can define a canonical twisted C-dynamical system (I,G, β |I , u |I). Moreover, I ×β,u G can be regarded as an ideal of A×β,u G. ∗This work is partially supported by Hong Kong RGC Direct Grant