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Journal of Operator Theory

Volume 87, Issue 1, Winter 2022  pp. 3-23.

Injective envelopes and the intersection property

Authors:  Rasmus Sylvester Bryder
Author institution: Department of Mathematics, University of Copenhagen, Copenhagen, 2100, Denmark

Summary: 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.

Keywords: reduced crossed product, injective envelope, primeness, intersection property, reduced group $C^*$-algebra

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