Injective envelopes and the intersection property

Abstract

We consider the ideal structure of a reduced crossed product of a unital $C^{\ast }$-algebra equipped with an action of a discrete group. More specifically we find sufficient and necessary conditions for the group action to have the intersection property, meaning that non-zero ideals in the reduced crossed product restrict to non-zero ideals in the underlying $C^{\ast }$-algebra. We show that the intersection property of a group action on a $C^{\ast }$-algebra is equivalent to the intersection property of the action on the equivariant injective envelope. We also show that the centre of the equivariant injective envelope always contains a $C^{\ast }$-algebraic copy of the equivariant injective envelope of the centre of the injective envelope. Finally, we give applications of these results in the case when the group is $C^{\ast }$-simple.

To appear in J. Operator Theory. 23 pages; v4; reorganised preliminaries and examples, restructured results