Injective envelopes and the intersection property

TL;DR

This paper investigates the conditions under which group actions on $C^*$-algebras have the intersection property, linking it to injective envelopes and providing applications for $C^*$-simple groups.

Contribution

It establishes the equivalence of the intersection property for group actions on $C^*$-algebras and their equivariant injective envelopes, advancing understanding of ideal structures.

Findings

The intersection property of a group action is equivalent to that on the equivariant injective envelope.

The center of the equivariant injective envelope contains a $C^*$-algebraic copy of the injective envelope of the center.

Applications are provided for the case when the group is $C^*$-simple.

Abstract

We consider the ideal structure of a reduced crossed product of a unital $C^*$-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 $C^*$-algebra. We show that the intersection property of a group action on a $C^*$-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 $C^*$-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 $C^*$-simple.