2024/06/27 by Vrej Zarikian, Zarikian, Vrej
Mathematics · #46L07 #46L55 #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #FOS: Mathematics #Limits and Structures in Graph Theory #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2406.19484
openalex publication_date 2024/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \unlhd 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 \rtimesα,rσG, 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) \rtimesα,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.