Unique Pseudo-expectations for C∗-inclusions

Given an inclusion D ⊆ C of unital C∗-algebras (with common unit), a unital completely positive linear map Φ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. Pseudo-expectations are generalizations of conditional expectations, but with the advantage that they always exist. The set PsExp(C,D) of all pseudoexpectations is a convex set, and when D is abelian, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its set of 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. In general, PsExp(C,D) is not a singleton. However there are large and natural classes of inclusions (e.g., when D is a regular MASA in C) such that there is a unique pseudo-expectation. Uniqueness of the pseudo-expectation typically implies interesting structural properties for the inclusion. For example, we show that when D ⊆ C ⊆ B(H) are von Neumann algebras, uniqueness of the pseudo-expectation implies that D′∩C is the center of D; moreover, when H is separable and D is abelian, we are able to characterize which von Neumann algebra inclusions have the unique pseudo-expectation property. For general inclusions of C∗-algebras with D abelian, we give a characterization of 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. As applications, we show that if an inclusion D ⊆ C has a unique pseudo-expectation Φ which is also faithful, then the C∗-envelope of any operator space X with D ⊆ X ⊆ C is the C∗-subalgebra of C generated by X; we also show that for many interesting classes of C∗-inclusions, having a faithful unique pseudo-expectation implies that D norms C, although this is not true in general. We provide a number of examples to illustrate the theory, and conclude with several unresolved questions.