Aperiodicity: The Almost Extension Property and Uniqueness of Pseudo-Expectations

Abstract

Abstract We prove implications among the conditions in the title for general $\text C^*$-inclusions $A\subseteq B$, 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 if the $\text C^*$-inclusion is aperiodic it has a unique pseudo-expectations, and if, in addition, this 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 $\text C^*$-algebras $C$ with $A\subseteq C \subseteq B$. If, in addition, $B$ is separable, then all these conditions are equivalent to the almost extension property.