Unique Pseudo-Expectations for \(C^\)-Inclusions

• Published in: arXiv.org 2015-08
• Main Authors: Pitts, David R, Zarikian, Vrej
• Format: Article
• Language: English
• Subjects: Algebra; Homomorphisms; Inclusions; Norms; Uniqueness
• ISSN: 2331-8422

Given an inclusion D \(\subseteq\) C of unital C*-algebras, a unital completely positive linear map \(\Phi\) of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D \(\subseteq\) C \(\subseteq\) B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' \(\cap\) C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D \(\subseteq\) C has a unique pseudo-expectation \(\Phi\) which is also faithful, then the C*-envelope of any operator space X with D \(\subseteq\) X \(\subseteq\) C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions.