Unique pseudo-expectations for hereditarily essential $C^*$-inclusions
Volume 283 / 2025
Abstract
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 $\{0\} \neq \mathcal J \lhd \mathcal C$, we have $\mathcal J \cap \mathcal A \neq \{0\}$. 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 the author, the $C^*$-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 $\mathcal A \rtimes _{\alpha ,r}^\sigma N \subseteq \mathcal A \rtimes _{\alpha ,r}^\sigma G$, for a discrete twisted $C^*$-dynamical system $(\mathcal A,G,\alpha ,\sigma )$ and a normal subgroup $N \lhd G$. 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 has a positive answer for regular hereditarily essential $C^*$-inclusions.
