Unique conditional expectations for abelian C⁎-inclusions

Let D⊆AD⊆A be an inclusion of unital abelian C⁎C⁎-algebras. In this note we characterize (in topological terms) when there is a unique conditional expectation E:A→DE:A→D, at least when AA is separable. We also provide the first example of an inclusion with a unique conditional expectation, but multiple pseudo-expectations (in the sense of Pitts).