Unique conditional expectations for abelian $C^{\ast }$-inclusions

Let $D\subseteq A$ be an inclusion of unital abelian $C^{\ast }$-algebras. In this note we characterize (in topological terms) when there is a unique conditional expectation $E:A\to D$, at least when $A$ is separable. As an application, we provide the first example of an inclusion with a unique conditional expectation, but multiple pseudo-expectations (in the sense of Pitts).