Topological treatments of some results on invariant ideals

Abstract

In this short article, we will give several results concerning closed invariant ideals of crossed products using some "topological arguments". Mathematics Subject Classification (2000): 46L05, 46L55Suppose that A is a separable C * -algebra, G is a second countable locally compact amenable group and α is an action of G on A. Let B := A × α G and Prim α A := t∈G\{e} Prim A \ (Prim A) t (where (Prim A) t is the set of elements in Prim A that are fixed byα t andα is the induced action of G on Prim A). By an invariant ideal of A, we mean a closed ideal that is invariant under α while an invariant ideal of B is a closed ideal that is invariant under the dual coactionα. Note that an ideal of B is invariant if and only if it is of the form I × α G for an invariant ideal I of A.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]). 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. Our main result is Theorem 7. Note that Theorem 7(a) can be regarded as a general form of [6, Theorem 9] because ker −1 (q G (Int(Prim α A))) = (0) when α is almost free (thus, this result give us a bit more information in the case when α is not almost free). On the other hand, notice that the assumption in Theorem 7(b) (i.e. Int(Prim α A) = ∅) is significantly weaker than almost freeness although in this case, we need to assume that a closed ideal J is essential in order to conclude that it contains an invariant ideal.For simplicity, we consider only ordinary actions in this article but the main theorem remains true for twisted actions because of the stabilisation trick of Packer-