Invariant ideals of twisted crossed products

Abstract

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).

Topological treatments of some results on invariant ideals

“…(a) As noted in the second paragraph of this article, Theoerm 7(a) implies [6,Theorem 9]. Note that in order to obtain Theoerm 7(a), one only needs Lemmas 1 and 4 as well as Proposition 6(a).…”

“…(c) Note that in Proposition 6(a), if Y 0 is both open and dense, then the only closed set C with ψ −1 (ϕ(C)) being dense in Y is the whole X. It is natural to ask if the same is true when Y 0 is not open (which will give a generalization of [6,Theorem 9]). This is true if X is Hausdorff.…”

“…In [6,Theorem 9], it was proved that if α is almost free (i.e. Int(Prim α A) is dense in Prim A; see [6,Definition 4]), then any non-zero closed ideal of B will contain a non-zero invariant ideal (this result is a crucial step toward one of the main results in [7]).…”

“…Int(Prim α A) is dense in Prim A; see [6,Definition 4]), then any non-zero closed ideal of B will contain a non-zero invariant ideal (this result is a crucial step toward one of the main results in [7]). The aims of this article are to give another proof of [6, Theorem 9] using certain "topological arguments" and to study invariant ideals under similar situations using this topological method.…”

Bounded trace C*-algebras and integrable actions

“…Instead of free actions we consider almost free actions which have recently been introduced by Leung and Ng ( [15]), and we discuss the hypothesis that the G-orbits are locally closed.…”