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

The C^*-inclusion A ⊆ B is said to be hereditarily essential if for every intermediate C^*-algebra A ⊆ C ⊆ B and every non-zero ideal \0\ ≠ J C, we have that J ∩ A ≠ \0\. That is, A detects ideals in every intermediate C^*-algebra A ⊆ C ⊆ B. By a result of Pitts and Zarikian, a unital C^*-inclusion A ⊆ B is hereditarily essential if and only if every pseudo-expectation θ:B → I(A) for A ⊆ 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 A ⊆ A α,r^σ G, for a twisted C^*-dynamical system (A,G,α,σ). On the other hand, we settle the general question negatively by exhibiting C^*-irreducible inclusions of the form Cᵣ^*(G) ⊆ C(X) α,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.