  Let $D \subseteq A$ be an inclusion of unital abelian $C^*$-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).

Operator Algebras