Unique Pseudo-Expectations for Hereditarily Essential $C^{\ast }$-Inclusions

Abstract

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

12 pages, comments welcome