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

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