Aperiodicity: the almost extension property and uniqueness of pseudo-expectations

Abstract

We prove implications among the conditions in the title for an inclusion of a C*-algebra A in a C*-algebra B, and we also relate this to several other properties in case B is a crossed product for an action of a group, inverse semigroup or an étale groupoid on A. We show that an aperiodic C*-inclusion has a unique pseudo-expectation. If, in addition, the unique pseudo-expectation is faithful, then A supports B in the sense of the Cuntz preorder. The almost extension property implies aperiodicity, and the converse holds if B is separable. A crossed product inclusion has the almost extension property if and only if the dual groupoid of the action is topologically principal. Topologically free actions are always aperiodic. If A is separable or of Type I, then topological freeness, aperiodicity and having a unique pseudo-expectation are equivalent to the condition that A detects ideals in all intermediate C*-algebras. If, in addition, B is separable, then all these conditions are equivalent to the almost extension property.