Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions

The $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is said to be hereditarily essential if for every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$ and every non-zero ideal $\{0\} \neq \mathcal{J} \unlhd \mathcal{C}$, we have that $\mathcal{J} \cap \mathcal{A} \neq \{0\}$. That is, $\mathcal{A}$ detects ideals in every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$. By a result of Pitts and Zarikian, a unital $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is hereditarily essential if and only if every pseudo-expectation $\theta:\mathcal{B} \to I(\mathcal{A})$ for $\mathcal{A} \subseteq \mathcal{B}$ is faithful. A decade-old open question asks whether hereditarily essential $C^*$-inclusions must have unique pseudo-expectations? In this note, we answer the question affirmatively for some important classes of $C^*$-inclusions, in particular those of the form $\mathcal{A} \subseteq \mathcal{A} \rtimes_{\alpha,r}^\sigma G$, for a twisted $C^*$-dynamical system $(\mathcal{A},G,\alpha,\sigma)$. On the other hand, we settle the general question negatively by exhibiting $C^*$-irreducible inclusions of the form $C_r^*(G) \subseteq C(X) \rtimes_{\alpha,r} G$ with multiple conditional expectations. Our results leave open the possibility that the question might have a positive answer for regular hereditarily essential $C^*$-inclusions.